Phase Transformations: Examples from Titanium and Zirconium Alloys (Pergamon Materials Series, Volume 12)

Phase Transformations Examples from Titanium and Zirconium Alloys PERGAMON MATERIALS SERIES Series Editor: Robert W...

Author: Srikumar Banerjee | Pradip Mukhopadhyay

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Phase Transformations Examples from Titanium and Zirconium Alloys

PERGAMON MATERIALS SERIES

Series Editor: Robert W. Cahn frs Department of Materials Science Cambridge, UK Vol. 1 Vol. 2 Vol. 3

and

Metallurgy,

University

of

Cambridge,

CALPHAD by N. Saunders and A. P. Miodownik Non-Equilibrium Processing of Materials edited by C. Suryanarayana Wettability at High Temperatures by N. Eustathopoulos, M. G. Nicholas and B. Drevet Vol. 4 Structural Biological Materials edited by M. Elices Vol. 5 The Coming of Materials Science by R. W. Cahn Vol. 6 Multinuclear Solid-State NMR of Inorganic Materials by K. J. D. MacKenzie and M. E. Smith Vol. 7 Underneath the Bragg Peaks: Structural Analysis of Complex Materials by T. Egami and S. J. L. Billinge Vol. 8 Thermally Activated Mechanisms in Crystal Plasticity by D. Caillard and J. L. Martin Vol. 9 The Local Chemical Analysis of Materials by J. W. Martin Vol. 10 Metastable Solids from Undercooled Melts by D. M. Herlach, P. Galenko and D. Holland-Moritz Vol. 11 Thermo-Mechanical Processing of Metallic Materials by B. Verlinden, J. Driver, I. Samajdar and R. D. Doherty

Phase Transformations Examples from Titanium and Zirconium Alloys

S. Banerjee and P. Mukhopadhyay Bhabha Atomic Research Centre, Mumbai, India

Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo Pergamon is an imprint of Elsevier

Pergamon is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2007 Copyright © 2007 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN 13: 978-0-08-042145-2 For information on all Pergamon publications visit our web site at books.elsevier.com Printed and bound in Great Britain 07 08 09 10

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This book is dedicated to the memory of Robert W. Cahn, who sadly died in April 2007.

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Contents Foreword Preface Acknowledgements

xvii xix xxi

CHAPTER 1 Phases and Crystal Structures Symbols and Abbreviations 1.1 Introduction 1.2 Polymorphism 1.3 Phase Diagrams of Elemental Titanium and Zirconium 1.3.1 Introductory remarks 1.3.2 Titanium 1.3.3 Zirconium 1.3.4 Epilogue 1.3.5 Phase stability and electronic structure 1.3.6 Some features of transition metals 1.4 Effect of Alloying 1.4.1 Introductory remarks 1.4.2 Alloy classification 1.4.3 Titanium alloys 1.4.4 Zirconium alloys 1.4.5 Stability of titanium and zirconium alloys 1.5 Binary Phase Diagrams 1.5.1 Introductory remarks 1.5.2 Ti–X systems 1.5.3 Zr–X systems 1.5.4 Representative examples of Ti–X and Zr–X phase diagrams 1.6 Non-Equilibrium Phases 1.6.1 Introductory remarks 1.6.2 Martensite phase 1.6.2.1 Crystallography 1.6.2.2 Transformation temperatures 1.6.2.3 Morphology and substructure vii

3 3 4 4 7 7 9 10 11 13 18 21 21 21 21 23 24 26 26 27 29 29 43 43 44 44 47 48

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1.6.3 Omega Phase 1.6.3.1 Athermal and isothermal 1.6.3.2 Crystallography 1.6.3.3 Morphology 1.6.3.4 Diffraction effects 1.6.4 Phase separation in -phase 1.7 Intermetallic Phases 1.7.1 Introductory remarks 1.7.2 Intermetallic phase structures: atomic layer stacking 1.7.3 Derivation of intermetallic phase structures from simple structures 1.7.4 Intermetallic phases with TCP structures in Ti–X and Zr–X systems 1.7.5 Phase stability in zirconia-based systems 1.7.5.1 ZrO2 polymorphs 1.7.5.2 Stabilization of high temperature polymorphs 1.7.5.3 ZrO2 –MgO system 1.7.5.4 ZrO2 –CaO system 1.7.5.5 ZrO2 –Y2 O3 system References Appendix CHAPTER 2 Classification of Phase Transformations Symbols and Abbreviations 2.1 Introduction 2.2 Basic Definitions 2.3 Classification Schemes 2.3.1 Classification based on thermodynamics 2.3.2 Classifications based on mechanisms 2.3.3 Classification based on kinetics 2.4 Syncretist Classification 2.5 Mixed Mode Transformations 2.5.1 Clustering and ordering 2.5.2 First-order and second-order ordering 2.5.3 Displacive and diffusional transformations 2.5.4 Kinetic coupling of diffusional and displacive transformations References

49 49 50 51 51 52 53 53 55 61 62 62 62 63 65 66 67 67 73

89 89 89 90 92 93 101 105 105 115 115 116 120 120 122

Contents

CHAPTER 3 Solidification, Vitrification, Crystallization and Formation of Quasicrystalline and Nanocrystalline Structures List of Symbols 3.1 Introduction 3.2 Solidification 3.2.1 Thermodynamics of solidification 3.2.2 Morphological stability of the liquid/solid interface 3.2.3 Post-solidification transformations 3.2.4 Macrosegregation and microsegregation in castings 3.2.5 Microstructure of weldments of Ti- and Zr-based alloys 3.3 Rapidly Solidified Crystalline Products 3.3.1 Extension of solid solubility 3.3.2 Dispersoid formation in rapidly solidified Ti alloys 3.3.3 Transformations in the solid state 3.4 Amorphous Metallic Alloys 3.4.1 Glass formation 3.4.2 Thermodynamic considerations 3.4.3 Kinetic considerations 3.4.4 Microstructures of partially crystalline alloys 3.4.5 Diffusion 3.4.6 Structural relaxation 3.4.7 Glass transition 3.5 Crystallization 3.5.1 Modes of crystallization 3.5.2 Crystallization in metal–metal glasses 3.5.3 Kinetics of crystallization 3.5.4 Crystallization kinetics in Zr 76 Fe1−x Nix 24 glasses 3.6 Bulk Metallic Glasses 3.7 Solid State Amorphization 3.7.1 Thermodynamics and kinetics 3.7.2 Amorphous phase formation by composition-induced destabilization of crystalline phases 3.7.3 Glass formation in diffusion couples 3.7.4 Amorphization by hydrogen charging 3.7.5 Glass formation in mechanically driven systems 3.7.6 Radiation-induced amorphization 3.8 Phase Stability in Thin Film Multilayers

ix

127 127 128 128 128 135 140 141 145 150 152 153 153 157 157 159 165 171 176 180 182 184 185 187 192 200 205 212 215 220 220 225 226 229 237

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3.9

Quasicrystalline Structures and Related Rational Approximants 3.9.1 Icosahedral phases in Ti- and Zr-based systems References CHAPTER 4 Martensitic Transformations Symbols and Abbreviations 4.1 Introduction 4.2 General Features of Martensitic Transformations 4.2.1 Thermodynamics 4.2.2 Crystallography 4.2.3 Kinetics 4.2.4 Summary 4.3 BCC to Orthohexagonal Martensitic Transformation in Alloys Based on Ti and Zr 4.3.1 Phase diagrams and Ms temperatures 4.3.2 Lattice correspondence 4.3.3 Crystallographic analysis 4.3.3.1 Morphology and substructure 4.3.3.2 Transition in morphology and substructure 4.3.4 Stress-assisted and strain-induced martensitic transformation 4.4 Strengthening due to Martensitic Transformation 4.4.1 Microscopic interactions 4.4.1.1 Lath boundaries 4.4.1.2 Twin boundaries and plate boundaries 4.4.2 Macroscopic flow behaviour 4.5 Martensitic Transformation in Ti–Ni Shape Memory Alloys 4.5.1 Transformation sequences 4.5.2 Crystallography of the B2 → R transformation 4.5.3 Crystallography of the B2 → B19 transformation 4.5.4 Crystallography of the B2 → B19 transformation 4.5.5 Self-accommodating morphology of Ni–Ti martensite plates 4.5.6 Shape memory effect 4.5.7 Reversion stress in a shape memory alloy 4.5.8 Thermal arrest memory effect 4.6 Tetragonal Monoclinic Transformation in Zirconia 4.6.1 Transformation characteristics 4.6.2 Orientation relation and lattice correspondence 4.6.3 Habit plane

241 248 252

259 259 260 261 261 266 277 280 281 282 289 294 304 320 324 326 329 329 331 335 339 340 342 342 345 347 352 356 360 362 362 363 366

Contents

4.7

Transformation Toughening of Partially Stabilized Zirconia (PSZ) 4.7.1 Crystallography of tetragonal → monoclinic transformation in small particles References CHAPTER 5 Ordering in Intermetallics List of Symbols 5.1 Introduction 5.2 Theoretical Treatments 5.2.1 Alloy phase stability 5.2.2 Order–disorder transformations 5.2.2.1 Historical developments 5.2.2.2 Static concentration wave model 5.2.2.3 Cluster variation method 5.2.3 The ground states of the Lenz and Ising model 5.2.4 Special point ordering 5.2.4.1 BCC special points 5.2.4.2 HCP special points 5.2.4.3 FCC special points 5.2.5 Concomitant clustering and ordering 5.2.6 A case study: Ti–Al system 5.3 Transformations in Ti3 Al-based alloys 5.3.1 → D019 ordering 5.3.2 Phase transformations in 2 -Ti3 Al-based systems 5.3.3 Structural relationships 5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63 /mmc) and ordered orthorhombic (Cmcm) phases 5.3.5 Transformation sequences 5.3.5.1 Transformation sequence in the alloy Ti–25 at.% Al–12.5 at.% Nb 5.3.5.2 Transformation sequence in the alloys Ti–25 at.% Al–25 at.% Nb, Ti–28 at.% Al–22 at.% Nb and Ti–24 at.% Al–15 at.% Nb 5.3.6 Phase reactions in Ti–Al–Nb system 5.4 Formation of Zr3 Al 5.4.1 Metastable Zr3 Al (D019 ) phase 5.4.2 Formation of the equilibrium Zr3 Al (L12 ) phase 5.4.3 +Zr2 Al → Zr3 Al peritectoid reaction

xi

369 372 373

379 379 380 383 384 386 387 389 392 397 401 404 406 407 407 412 416 416 417 421

424 428 430

431 432 436 437 439 441

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5.5

Phase Transformation in -TiAl-Based Systems 5.5.1 Structural relationship between 2 - and -phases 5.5.2 Phase reactions 5.5.2.1 Ti-34-37 at.% Al; → → 2 5.5.2.2 Ti-38-40 at.% Al; → 2 → 2 + 5.5.2.3 Ti-41-48 at.% Al; → + → 2 + 5.5.2.4 Ti-49-50 at.% Al; → 5.5.2.5 Ti49-50 at.% Al; → 5.5.3 Transformation mechanisms 5.5.3.1 Formation of the 2 + lamellar microstructure 5.5.3.2 Mechanism of the → massive transformation 5.5.3.3 Discontinuous coarsening of the lamellar 2 + microstructure 5.6 Site Occupancies in Ordered Ternary Alloys 5.6.1 Ordering tie lines 5.6.2 Kinetic modelling of B2 ordering in a ternary system 5.6.3 Influence of binary interaction parameters 5.6.4 B2 ordering in the Nb–Ti–Al system References CHAPTER 6 Transformations Related To Omega Structures List of Symbols 6.1 Introduction 6.2 Occurrence of the -Phase 6.2.1 Thermally induced formation of the -phase 6.2.2 Formation of equilibrium -phase under high pressures 6.2.3 Combined effect of alloying elements and pressure in inducing -transition 6.3 Crystallography 6.3.1 The structure of the -phase 6.3.2 The – lattice correspondence 6.3.3 The – lattice correspondence 6.4 Kinetics of the → Transformation 6.4.1 Athermal → transition 6.4.2 Thermally activated precipitation of the -phase 6.4.3 Pressure-induced → transformation 6.5 Diffuse Scattering

443 443 446 446 447 448 450 450 451 451 453 456 458 458 460 462 464 465

473 473 474 475 475 479 481 484 484 485 488 490 491 492 494 495

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6.6

Mechanisms of -Transformations 6.6.1 Lattice collapse mechanism for the → transformation 6.6.2 Formation of the plate-shaped induced by shock pressure in -alloys 6.6.3 Calculated total energy as a function of displacement 6.6.4 Incommensurate -structures 6.6.5 Stability of -phase and d-band occupancy 6.7 Ordered -Structures 6.7.1 Structural descriptions 6.7.2 Transformation sequences in Zr base alloys 6.7.3 Transformation sequences in Ti base alloys 6.7.4 Ordered -structures in other systems 6.7.5 Symmetry tree 6.8 Influence of -Phase on Mechanical Properties 6.8.1 Hardening and embrittlement due to -phase 6.8.2 Dynamic strain ageing due to -precipitation References CHAPTER 7 Diffusional Transformations List of Symbols 7.1 Introduction 7.2 Diffusion 7.2.1 Diffusion mechanisms 7.2.2 Flux equations: Fick’s laws 7.2.3 Self- and tracer-diffusion coefficients in -Zr and -Ti 7.2.4 Self- and tracer-diffusion coefficients in -Zr and -Ti 7.2.5 Interdiffusion 7.2.6 Phase formation in chemical diffusion 7.2.6.1 Phase nucleation 7.2.6.2 Phase growth 7.2.7 Diffusion bonding 7.3 Phase Separation 7.3.1 Phase separation mechanisms 7.3.2 Analysis of a phase diagram showing a miscibility gap 7.3.3 Microstructural evolution during phase separation in the -phase 7.3.4 Monotectoid reaction – a consequence of -phase immiscibility

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499 499 504 506 509 516 518 518 522 530 533 534 536 536 539 550

557 557 558 560 560 562 564 566 570 578 580 581 584 587 589 597 603 606

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7.3.5 Precipitation of -phase in supersaturated -phase during tempering of martensite 7.3.6 Decomposition of orthorhombic -martensite during tempering 7.3.7 Phase separation in -phase as precursor to precipitation of - and -phases 7.4 Massive Transformations 7.4.1 Thermodynamics of massive transformations 7.4.2 Massive transformations in Ti alloys 7.5 Precipitation of -Phase in -Matrix 7.5.1 Morphology 7.5.2 Orientation relation 7.5.3 Invariant line strain condition 7.5.4 Interfacial structure and growth mechanisms 7.5.5 Morphological evolution in mesoscale 7.6 Precipitation of Intermetallic Phases 7.6.1 Precipitation of intermetallic compounds from dilute solid solutions 7.6.2 Precipitation in ordered intermetallics: transformation of 2 -phase to O-phase 7.7 Eutectoid Decomposition 7.7.1 Active eutectoid systems 7.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe system 7.8 Microstructural Evolution During Thermo-Mechanical Processing of Ti- and Zr-based Alloys 7.8.1 Identification of hot deformation mechanisms through processing maps 7.8.2 Development of microstructure during hot working of Ti alloys 7.8.2.1 -alloys 7.8.2.2 + alloys 7.8.2.3 -alloys 7.8.2.4 Ti-aluminides 7.8.3 Hot working of Zr alloys 7.8.3.1 and near--Zr alloys 7.8.3.2 + alloys 7.8.3.3 -alloys 7.8.4 Development of texture during cold working of Zr alloys 7.8.5 Evolution of microstructure during fabrication of Zr–2.5 wt% Nb alloy tubes References

609 616 618 623 623 626 632 633 642 643 648 655 657 657 662 670 675 676 683 684 687 687 687 689 690 691 692 696 701 701 706 710

Contents

CHAPTER 8 Interstitial Ordering List of Symbols 8.1 Introduction 8.2 Hydrogen In Metals 8.2.1 Ti–H and Zr–H phase diagrams 8.2.2 Terminal solid solubility 8.3 Crystallography and Mechanism of Hydride Formation 8.3.1 Formation of -hydride in the - and -phases 8.3.2 Lattice correspondence of -, - and -phases 8.3.3 Crystallography of → transformation 8.3.4 Crystallography of → transformation 8.3.5 Mechanism of the formation of -hydrides 8.3.6 Hydride precipitation in the / interface 8.3.7 Formation of -hydride 8.4 Hydrogen-Related Degradation Processes 8.4.1 Uniform hydride precipitation 8.4.2 Hydrogen Migration 8.4.3 Stress reorientation of hydride precipitates 8.4.4 Delayed hydride cracking 8.4.5 Formation of hydride blisters 8.5 Thermochemical Processing of Ti Alloys by Temporary Alloying With Hydrogen 8.6 Hydrogen Storage In Intermetallic Phases 8.6.1 Laves phase compounds 8.6.2 Thermodynamics 8.6.3 Ti- and Zr-based hydrogen storage materials 8.6.3.1 Ti-based hydrogen storage materials 8.6.3.2 Zr-based hydrogen storage materials 8.6.4 Applications 8.7 Oxygen Ordering In -Alloys 8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O 8.7.2 Oxidation kinetics and mechanism 8.8 Phase transformations in Ti-rich end of the Ti–N system References

xv

719 719 720 721 722 725 728 728 729 730 735 737 737 739 741 742 743 745 746 747 753 754 754 756 756 758 759 761 764 764 769 772 780

CHAPTER 9 Epilogue References

785 800

Index

801


Foreword The present volume looks at phase transformations essentially from a physical metallurgist’s view point, in consonance with the background and the research experience of the authors, and has some distinguishing features. Some, though not all, of these are enumerated in the following. Almost all types of phase transformations and reactions that are commonly encountered in inorganic materials, such as alloys, intermetallics and ceramics, have been covered and the underlying thermodynamic, kinetic and crystallographic aspects elucidated. It has generally been customary in metallurgical literature to draw examples from iron-based alloys for describing the characteristic features of different types of transformations, in view of the wide variety of transformations occurring in these alloys. The authors of this monograph have cited examples of all the phase transformations and reactions discussed from titanium- and zirconium-based systems and have successfully demonstrated that these alloys, intermetallics and ceramics exhibit an even wider range of phase changes as compared to ferrous systems and that the simpler crystallography involved renders them more suitable for developing a basic understanding of the transformations. Phase transformations are brought about due to changes in external constraints which include thermodynamic variables such as temperature and pressure. Till recently, the emphasis in metallurgical literature has been on the delineation of temperature-induced transformations. In this book, transformations driven by pressure changes, radiation and deformation and those occurring in nanoscale multilayers have also been brought to the fore, while accepting the pre-eminent position occupied by the temperature-induced ones. Order–disorder transformations, many of which constitute very good examples of continuous transformations, have been dealt with in a comprehensive manner. It has been demonstrated that first principles calculations of phase stability can yield meaningful results, consistent with experimental observations. Displacive transformations, both shear dominated (martensite, shock pressure induced omega) and shuffle dominated (omega), have been covered in a cogent manner. Some crystallographic bcc to hcp transformations, which occur by diffusional as well as by displacive modes, have been identified, compared and contrasted, in terms of the experimentally observable features which characterize them. The authors, who have a lifetime of experience in investigating phase transformations in zirconium and titanium alloys, have handled an ambitious project xvii

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by trying to bring diverse topics under the same cover. And they have certainly not failed in their endeavor. One could always point out that non-metallic systems have not been adequately represented in their treatment. However, in quite a few instances, they have compared phase transformations occurring in alloys, intermetallics and ceramics and have demonstrated that the underlying principles pertaining to all these systems are basically the same. The multidisciplinary and interdisciplinary interest in the area of phase changes have engendered a variety of approaches with regard to the study of phase transformations, each exhibiting some distinctive features. Physicists are interested primarily in the motivation or, in other words, the why of a transformation. They concern themselves mainly with higher order, continuous phase transitions occurring in simple, composition-invariant systems. Chemists, metallurgists and ceramists, by contrast, focus a major part of their attention on phase transformations (and phase reactions) involving alterations in crystal structure, chemical composition and state of order. Of great concern to metallurgists are the mechanisms, or the how, of such transformations. Phase changes of interest to geologists are similar to those encountered in metallic and ceramic systems but generally take place over much more extended temporal and spatial scales under extreme conditions of temperature and pressure. The present volume will be useful to students, research workers and professionals belonging to all these disciplines. In my judgment, the authors of this volume have done a commendable job while addressing phase transformations and phase reactions, drawing apposite examples from titanium-and zirconium-based systems, and have been able to produce a monograph which was not there but which should very much have been there. I congratulate them on this count. C.N.R. Rao, F.R.S. Linus Pauling Research Professor, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India

Preface Studies on phase transformations in metallic materials form a major part of physical metallurgy. The terms phase transitions and phase transformations are often used in an interchangeable manner in metallurgical literature although it is realized that the former generally refers to transitions between two phases having the same chemical composition while the latter spans a wider range of phenomena, including phase reactions leading to compositional changes. Having made this distinction, we would like to mention at the outset that the present volume deals with phase transformations. We started our respective research careers almost four decades back by looking into some phase transformations and phase reactions occurring in zirconium alloys. As we gathered more and more experience, we realized that these alloys, together with titanium alloys, exhibit nearly all types of phase transformations encountered in inorganic materials and that in this respect these are more versatile than even ferrous alloys. Moreover, the crystallographic features associated with the phase changes are often simpler in these systems, making them more suitable for providing a basic understanding of the relevant phenomena. In earlier days, some of the important issues in the area of phase transformations in alloys, intermetallics and ceramics pertained to the following: (1) crystallographic aspects of martensitic transformations, including the role of the lattice-invariant shear, in determining martensite morphology and substructure and the strengthening contribution of the latter; (2) distinguishing features of diffusional and displacive transformations and mechanisms of hybrid transformations; (3) analysis and synthesis of phase diagrams and the prediction of the sequence of phase transformations on the basis of phase diagram analyses; (4) spinodal decomposition leading to a homogenous phase separation process and the evolution of microstructure in systems exhibiting instability in respect of concentration and/or displacement waves of short and/or long wavelengths; (5) driving force, kinetics and mechanisms relevant to displacive phase transformations and the role of strain fluctuations and their localization in the nucleation of such transformations; (6) formation of amorphous structure in metallic materials, stability of the amorphous phase and the modes of crystallization on appropriate processing; (7) the effect of factors such as pressure, deformation and radiation on phase transformations. xix

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A number of research groups all over the world, including our group, responded to the challenges thrown by these issues. The background was well set as the information and knowledge accumulated on the basis of metallography observations (mainly at light microscopy levels), X-ray diffraction results and studies on kinetics had already provided a fair understanding of the mechanisms of different types of phase transformations. Theoretical developments such as the phenomenological theory of martensite crystallography, the thermodynamical theory of spinodal decomposition and the theory of the growth kinetics of precipitates had had noteworthy success in making quantitative predictions regarding many an aspect of solid state phase transformations. That was also the time when transmission electron microscopy emerged as a powerful technique for making observations, morphological as well as crystallographic, at a much higher resolution than hitherto available, enabling physical metallurgists to resolve a number of mechanism-related problems which had been raised on the basis of theoretical and experimental investigations carried out earlier. We are happy to state that each of the issues listed above has been addressed, in some manner or the other, by our colleagues and by us over the years to enhance our understanding and appreciation of these. If one scans today’s literature on phase transformations, one will find that most of these issues, though better comprehended than before, continue to be in the limelight. However, the experimental tools now available have enormously improved our ability to study phenomena at much higher levels of spatial as well as temporal resolution. This superior experimental capability, supplemented by tremendously enhanced computing power, is providing a much better understanding of phase transformation phenomena. We do hope that the readers of this volume will get a flavour of these advancements. The book is divided into nine chapters. The first of these provides some sort of an introduction to the various types of phase changes covered later on. The second chapter delineates different schemes of classification of phase transformations in a general manner. The following six chapters deal with specific types of transformations. An attempt has been made to elucidate the basic principles pertaining to the relevant transformations, in general terms, at the beginning of each of these chapters because we have felt that this would be pedagogically advantageous for developing a clear understanding of the subject. However, we have taken care to ensure that all the illustrative examples are drawn from titaniumand zirconium-based systems. The final chapter is in the nature of an epilogue. Srikumar Banerjee Pradip Mukhopadhyay

Acknowledgements This book reflects the totality of the experience gained by us during our research career which, in the formative years, was under the guidance of R. Krishnan in Metallurgy Division, Bhabha Atomic Research Centre (BARC). Our research has been almost entirely supported by this institute (BARC), where a sustained activity on the physical metallurgy of zirconium has remained in focus for nearly four decades. It is here that we have had the benefit of interacting with M.K. Asundi, V.S. Arunachalam, P. Rodriguez, B.D. Sharma, R. Chidambaram, C.V. Sundaram and C.K. Gupta over the years. Interactions with other major centres of physical metallurgy research in the country have also been of considerable help. In this connection, we would like to especially acknowledge the fruitful discussions on many aspects of phase transformations research with P.R. Dhar of Indian Institute of Technology (IIT), Kharagpur; S. Ranganathan and K. Chattopadhyay of Indian Institute of Science (IISc), Bangalore; T.R. Anantharaman, P. Rama Rao, P. Ramachandrarao and S. Lele of Banaras Hindu University (BHU), Varanasi; D. Banerjee and K. Muraleedharan of Defence Metallurgical Research Laboratory (DMRL), Hyderabad; and V.S. Raghunathan of Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam. One of us (P. Mukhopadhyay) was introduced to ordering reactions in titanium aluminides by P.R. Swann at Imperial College, London, while the other (S. Banerjee) has had productive collaborations with R.W. Cahn and B. Cantor at University of Sussex, Brighton; M. Wilkens and K. Urban at Institut für Physik, Max-Planck Institut für Metallforschung, Stuttgart; and H.L. Fraser, R. Banerjee and J.C. Williams at the Ohio State University, Columbus. We also have had several occasions to imbibe pertinent ideas from H.I. Aaronson of Carngie-Mellon University, Pittsburgh; J.W. Cahn and L.A. Bendersky of National Institute of Standards and Technologies (NIST), Washington, D.C.; J.W. Christian of University of Oxford and V.K. Vasudevan of University of Cincinatti. We must acknowledge our indebtedness to the authors of many of the publications which have been instrumental in nurturing our understanding of the topics covered in this book. We have been extremely fortunate in having a continuous stream of bright colleagues in the course of our professional career. They have perhaps given us much more in terms of ideas and concepts than whatever advice and guidance we have been able to offer. We take this opportunity to list the names of some of those colleagues in the approximate sequence of our coming in contact with xxi

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them: S.J. Vijayakar, G.E. Prasad, L. Kumar, V. Seetharaman, E.S.K. Menon, M. Sundararaman, V. Raman, R. Kishore, U.D. Kulkarni, J.K. Chakravartty, G.K. Dey, K. Madangopal, D. Srivastava, R. Tewari, A.K. Arya, R.V. Ramanujam, J.B. Singh. Needless to say, this list is, by no means, complete. During the preparation of the manuscript of this book we received substantial help from many of our colleagues, notably G.K. Dey, D. Srivastava, A.K. Arya, R. Tewari, A. Laik, G.B. Kale, K. Bhanumurthy, R.N. Singh, S. Ramanathan and J.K. Chakravartty. We have received sustained secretarial assistance from M. Ayyappan and P. Khattar. P.B. Khedkar and A. Agashe have been mainly responsible for preparing the illustrations. We are grateful to Elsevier Publishers for their patience and readiness to help. Above all, we are greatly indebted to Robert Cahn, whose constant encouragement and occasional reprimands have contributed considerably to the completion of this work. He passed away at a time when this volume was in the proof-setting stage. His death has indeed created a void in the physical metallurgy community that will take a long time to be filled. To us it has been an irreparable loss, professional and personal. We dedicate this book to the memory of our parents and of Prof. Robert W. Cahn. Srikumar Banerjee Pradip Mukhopadhyay

Chapter 1

Phases and Crystal Structures 1.1 Introduction 1.2 Polymorphism 1.3 Phase Diagrams of Elemental Titanium and Zirconium 1.3.1 Introductory remarks 1.3.2 Titanium 1.3.3 Zirconium 1.3.4 Epilogue 1.3.5 Phase stability and electronic structure 1.3.6 Some features of transition metals 1.4 Effect of Alloying 1.4.1 Introductory remarks 1.4.2 Alloy classification 1.4.3 Titanium alloys 1.4.4 Zirconium alloys 1.4.5 Stability of titanium and zirconium alloys 1.5 Binary Phase Diagrams 1.5.1 Introductory remarks 1.5.2 Ti–X systems 1.5.3 Zr–X systems 1.5.4 Representative examples of Ti–X and Zr–X phase diagrams 1.6 Non-Equilibrium Phases 1.6.1 Introductory remarks 1.6.2 Martensite phase 1.6.3 Omega phase 1.6.4 Phase separation in -phase 1.7 Intermetallic Phases 1.7.1 Introductory remarks 1.7.2 Intermetallic phase structures: atomic layer stacking 1.7.3 Derivation of intermetallic phase structures from simple structures 1.7.4 Intermetallic phases with TCP structures in Ti–X and Zr–X systems 1.7.5 Phase stability in zirconia-based systems References Appendix

4 4 7 7 9 10 11 13 18 21 21 21 21 23 24 26 26 27 29 29 43 43 44 49 52 53 53 55 61 62 62 67 73


Chapter 1

Phases and Crystal Structures Symbols A Cij C Cp e/a G: H P S T V Va p s ij bcc: fcc: hcp: -phase: -phase: m Ms Mf s s To AIP:

and Abbreviations Elastic anisotropy ratio (A = C44 /C ) Elastic stiffness modulus (elastic constant) Elastic shear stiffness modulus; shear constant; (C = C11 − C12 /2) Specific heat at constant pressure Electron to atom ratio Gibbs free energy (G = H − TS) Enthalpy Pressure Entropy Temperature Volume Atomic volume Piston velocity Shock velocity Thermodynamic interaction parameter between elements i and j Body centred cubic Face centred cubic Hexagonal close packed hcp phase in Ti- and Zr-based alloys bcc phase in Ti- and Zr-based alloys hcp martensite Orthorhombic martensite Generic martensite ( or ) Temperature at which martensite starts forming during quenching Temperature at which martensite formation is completed during quenching Temperature at which the m → reversion starts on up-quenching Temperature at which athermal phase starts forming during quenching Temperature at which the free energies of the parent () and product (m ) phases are equal. Ab initio pseudopotential 3

4

Phase Transformations: Titanium and Zirconium Alloys

ASA: ASW: FPLAPW: LAPW: LCGTO: LDA: LMTO: MC: MD: MT: NFE: QMC: QSD: TB:

1.1

Atomic sphere approximation Augmented spherical wave Full potential linear augmented plane wave Linear augmented plane wave Linear combination of gaussian type orbitals Local density approximation Linear muffin tin orbital Monte carlo Molecular dynamics Muffin tin Nearly free electron Quantum monte carlo Quantum structural diagram Tight binding

INTRODUCTION

Titanium (Ti), zirconium (Zr) and hafnium (Hf) are transition metals belonging to Group 4 (nomenclature as per the recommendations of IUPAC 1988) of the periodic table of elements. The interest in the metals Ti and Zr and in alloys based on them has gained momentum from the late 1940s in view of their suitability for being used as structural materials in certain rapidly developing industries; particularly, the aerospace and chemical industries in the case of Ti alloys and the nuclear power industry in the case of Zr alloys. Some important characteristics of these metals are listed in Table 1.1. It can be seen from this table that the electronic ground state configurations of these metals are Ar3d2 4s2 and Kr4d2 5s2 , respectively. The similarity in the dispositions of the outer electrons, i.e. the four electrons (two s electrons and two d electrons) outside the inert gas shells (M shell for Ti and N shell for Zr) is, to a large extent, responsible for the similarities in some of the chemical and physical properties of these two metals and as a corollary, in many aspects of their chemical and physical metallurgy, including alloying behaviour.

1.2

POLYMORPHISM

Apart from existing in solid, liquid and gaseous states, many elements exhibit a special feature: they adopt different crystal structures in the solid state under different conditions of temperature or pressure or external field. The transition from

Phases and Crystal Structures

5

Table 1.1. Some characteristics of elemental Ti and Zr. Property

Element Ti

Atomic number (Z) Number of naturally occurring isotopes Atomic weight Electronic ground state configuration Density at 298 K kg/m3 Melting temperature (K) Boiling temperature (K) Enthalpy of fusion ( Hf ) kJ/mol Electronegativity Metal radius (nm)

22 5 47.90

Ar 3d2 4s2 4510 1941 3533 16.7 1.5 0.147

Zr 40 5 91.22

Kr 4d2 5s2 6510 2128 4650 18.8 1.4 0.160

References: Froes et al. 1996, Kubaschewski et al. 1993, McAuliffe and Bricklebank 1994, Soloveichik 1994.

one modification (allotrope) to another is termed a polymorphous transformation or a phase transformation (transition). A phase transition is associated with changes in structural parameters and/or in the ordering of electron spins (Steurer 1996). It will be discussed in a later chapter that two basically different types of phase transitions may be encountered: first-order transitions and second-order (or higher order) transitions. A transition of the former type is associated with discontinuous changes in the first derivatives of the Gibbs free energy, G = H − TS, while a transition of the latter type is characterized by discontinuous changes in the second (or higher order) derivatives of the Gibbs free energy and there are no jumpwise changes in the first derivatives. In either type of transition, the crystal structure undergoes a discontinuous change at the transition point (e.g. transition temperature or transition pressure). It is not necessary to have a symmetry relationship between the parent and the product phases in a first-order transition. However, in a second-order transition a group/subgroup relationship can always be found in relation to the symmetry groups associated with the crystal structures of the two phases. Elemental Ti and Zr (and Hf) exhibit temperature induced as well as pressure induced polymorphism. The pertinent phases, transition temperatures and transition pressures are listed in Table 1.2 and Table 1.3. It can be seen from Table 1.2 that for both Ti and Zr, the high temperature phase, termed the -phase, has the relatively “open” bcc structure while the low temperature phase, termed the -phase, has the close packed hcp structure. The hcp structure of the -phase is, however, slightly compressed in the sense that the value of the axial ratio is smaller than the ideal value of 1.63. It has been pointed out (McQuillan 1963, Collings 1984) that the more open bcc structure has a higher vibrational entropy as compared to

6


Table 1.2. Allotropic forms of elemental Ti and Zr at atmospheric pressure (Massalski et al. 1992) (Variable: temperature). Element

Phase

Temperature regime (K)

Enthalpy of transformation (kJ/mol)

Crystal structure

Ti

Alpha() Beta()

Up to 1155 1155–1943

4 174 2

hexagonal close packed body centred cubic

Zr

Alpha() Beta()

Up to 1139 (1136) 1139–2128

4 1033 9

hexagonal close packed body centred cubic

Note: The figures in parentheses are from Kubaschewski et al. 1993.

Table 1.3. Allotropic forms of elemental Ti and Zr at room temperature (Steurer 1996) (Variable: pressure) Element

Phase

Pressure regime (GPa)

Crystal structure

Ti

Alpha() Omega( )

Up to 2 >2

hexagonal close packed hexagonal

Zr

Alpha() Omega( ) Omega prime ( )

Up to 2 2–30 > 30

hexagonal close packed hexagonal body centred cubic

the close packed hcp structure and as a consequence of this, the free energy of a competing bcc lattice will decrease more rapidly than that of the hcp lattice with increasing temperature; a temperature will ultimately be reached at which the free energy of the former will be less than that of the latter so that the bcc form will be more stable. The -phase can be obtained from the -phase by the application of sufficient pressure in elemental Ti and Zr. Some crystallographic data pertaining to all these phases are presented in Table 1.4. The structure of the -phase has been determined to be either hexagonal, belonging to the space group P6/mmm ¯ (Silcock 1958), or trigonal, belonging to the space group P 3m1 (Bagariatskii et al. 1959), depending on the solute concentration. The equivalent positions in the unit cell of the structure are 000; 2/3 1/3 (1/2 − z); 1/3 2/3 (1/2 + z). For the ideal structure with hexagonal (P6/mmm) symmetry, z = 0 while 0 < z < 1/6 defines ¯ a non-ideal structure with trigonal (P 3m1) symmetry. There are three atoms in the unit cell. The axial ratio is close to 3/81/2 . The symmetry of the structure is high and as in the case of the simple hexagonal lattice, there are 24 point group operations (Ho et al. 1982). The packing density ( 0 57) associated with the hexagonal (hP3) structure of the -phase is lower than that for the bcc ( 0 68) and the hcp ( 0 74) structures. The occurrence of such an open structure in metals


7

Table 1.4. Crystal structures and lattice parameters of allotropic forms of elemental Ti and Zr (Massalski et al. 1992, Steurer 1996). Element

Ph

Crystal structure P

SN

PS

SG

Ti Va nm3 = 17 65 × 10−3

-Ti

Mg

A3

hP2

P63 /mmc

-Ti -Ti

W -Ti

A2 −

cI2 hP3

¯ Im3m P6/mmm

Zr Va nm3 = 23 28 × 10−3

-Zr

Mg

A3

hP2

P63 /mmc

-Zr -Zr

W -Ti

A2 −

cI2 hP3

¯ Im3m P6/mmm

W

A2

cI2

¯ Im3m

Lattice paramaters (nm)

Axial ratio

a = 0 29506 c = 0 46835 a = 0 33065 a = 0 4625 c = 0 2813

1 5873

a = 0 32316 c = 0 51475 a = 0 36090 a = 0 5036 c = 0 3109 −

1 0 0 6082 1 5929 1 0 0 617 −

Ph – Phase; P – Prototype structure; SN – Strukturbericht notation; PS – Pearson symbol; SG – Space group. Notes: 1. The lattice parameter values of - and - phase correspond to a temperature of 298 K. 2. The quantity Va refers to the atomic volume under ambient conditions.

with metallic d-bonding is somewhat unusual. Normally, the transition metals have close packed (fcc, hcp) or fairly close packed (bcc) structures. Open structures are common among the p-electron systems or the actinide elements (Duthie and Pettifor 1977, Skriver 1985). The stability of this phase has been attributed to the covalent bonding contribution from s-d electron transfer (Steurer 1996). In the case of Zr (and Hf), it has been found that on the application of substantially higher pressures (Table 1.3) the -phase transforms to the -phase, which has the bcc structure. Although a similar transformation has not been observed in the case of Ti, even at a pressure as high as 87 GPa, theoretical considerations indicate that this metal too would undergo such a transformation at still higher pressures (Ahuja et al. 1993, Steurer 1996). This issue is addressed in greater detail in a later chapter of this volume.

1.3

PHASE DIAGRAMS OF ELEMENTAL TITANIUM AND ZIRCONIUM

1.3.1 Introductory remarks From the point of view of the phase rule, a pure element represents a single component system which may exhibit different phases. The phase rule imposes

8


the condition f + p = c + 2, where f is the number of degrees of freedom in the pressure–temperature–composition space, p is the number of phases and c the number of components. For an element under temperature and pressure conditions of interest, f = 3 − p. This implies that a single phase is represented by an area in the pressure–temperature plane (p = 1 f = 2), a two-phase mixture is represented by a curve (p = 2 f = 1), which may be termed a phase boundary or phase line, and a three-phase mixture by a point (p = 3 f = 0), generally known as a triple point. A single component phase diagram is essentially a plot of areas representing phases, which are demarcated by phase boundaries, in the pressure–temperature or the P–T plane. A typical phase diagram of an element will generally show a vapour phase, a liquid phase and one or more solid phases. The phase boundaries have to abide by a few thermodynamic rules. The entropy change ( S) and the volume change ( V) across a phase boundary are related to the slope of the boundary by the Clausius–Clapeyron equation: dP S = dT V

(1.1)

This slope can be positive or negative: S must be positive for increasing temperature by the second law of thermodynamics, but V can be either positive or negative. d2 P The second derivative, dT 2 , gives a measure of the curvature and can be expressed as (Partington 1957): 2 d2 P 1 d V dP d V dP Cp =− − (1.2) +2 dT 2 V dP dT dT dT T For relatively incompressible solids like the transition metals, the terms on the right-hand side are small with the result that the phase boundaries have very small curvature and look like straight lines over the experimentally available pressure ranges (Young 1991). Experimental work on pressure-induced phase transformations in transition metals has been somewhat limited because of their low compressibility; phase changes may occur only at very high pressures which are difficult to achieve. Shock wave experiments are at present the most effective means of studying the high-pressure phase diagrams of these metals (Young 1991). A shock wave is a disturbance propagating at a supersonic speed in the medium. One can imagine the shock to be arising from a piston which moves into the medium at a constant velocity p . The boundary between the compressed and the uncompressed material will move ahead of the piston with a certain velocity s , which is termed the shock velocity. The basic objective of shock wave experiments is to measure the velocities p and s


9

and to determine from them the thermodynamic state of the host material. For most materials, p and s bear a linear relationship. But at a phase boundary this relationship may break down and the s versus p plot may show a discontinuity (McQueen et al. 1970). The reason for this is that a steady shock wave needs a sound speed that increases with compression and that this requirement is violated by a firstorder phase transition, with the result that the shock wave breaks up into a lowpressure wave (representing the untransformed material) and a high-pressure wave (representing the transformed material). The detectors register the arrival of only the first (i.e. low-pressure wave) and the two-wave region appears as a flat segment of constant s on the s versus p plot; a third segment appears on the plot when the shock velocity in the transformed material exceeds that corresponding to the untransformed material (Young 1991). The appearance of discontinuities in the s –p plane is thus a good indication of the occurrence of a first-order phase transition. 1.3.2 Titanium As stated earlier in this chapter, elemental Ti exists as the hcp -phase at room temperature under atmospheric pressure. On raising the pressure, while keeping the temperature constant, Ti transforms to the hexagonal -phase at around 2 GPa pressure. The – phase boundary has been reported to have a negative slope (Zilbershteyn et al. 1975, Vohra et al. 1982). This transition is associated with a large hysteresis and the equilibrium phase boundary has not been determined accurately (Young 1991). Further compression at room temperature to pressures upto 87 GPa has not shown any phase other than the -phase until recently (Xia et al. 1990a,b). As indicated earlier, this point will be covered in a subsequent chapter. Under atmospheric pressure, the -phase transforms to the denser -phase (bcc) at 1155 K. The – phase boundary has been determined by high temperature, static pressure measurements (Bundy 1963, Jayaraman et al. 1963). The triple point at which the -, - and -phases meet occurs at about 9.0 GPa and 940 K (Young 1991). The – phase boundary has been experimentally determined upto a pressure of 15 GPa (Bundy 1963). No phase other than the -, - and –phases has been found in Ti. Shock wave experiments conducted on elemental Ti have shown a discontinuity in the s –p curve; it has been suggested that this may correspond to the – or – transition (McQueen et al. 1970, Kutsar et al. 1982, Kiselev and Falkov 1982). The experimentally determined pressure–temperature phase diagram of Ti is shown in Figure 1.1 (Young 1991). Linear muffin tin orbital (LMTO) calculations which take into consideration the hcp, bcc, and fcc structures have predicted the stability of the -phase for pressures up to 30 GPa (Gyanchandani et al. 1990). The disposition of the – boundary (Figure 1.1) is not inconsistent with the theoretical prediction that at 0 K the -phase is the equilibrium phase in the case of Ti.

10

Phase Transformations: Titanium and Zirconium Alloys Ti 2.5 Liquid

T ( × 103 K)

2.0

bcc ( β)

1.5

1.0 hcp (α)

hex ( ω)

0.5

0

0

6

12

18

P (GPa)

Figure 1.1. Experimentally determined temperature–pressure phase diagram for Ti.

1.3.3 Zirconium As in the case of Ti, elemental Zr exists as the hcp -phase at room temperature and pressure, while on pressurization at this temperature it gets converted to the hexagonal -phase at a pressure of about 2 GPa. In this case also, the – phase line exhibits a negative slope (Jayaraman et al. 1963, Zilbershteyn et al. 1975, Guillermet 1987). A precise determination of the equilibrium transition pressure has, however, not been possible due to the occurrence of hysteresis (Young 1991). Static pressure experiments at room temperature have established that the -phase transforms to a bcc phase ( ) at a pressure of 30 GPa (Xia et al. 1990a,b). This bcc phase has been found to be the same as the -phase. It has been mentioned earlier that under atmospheric pressure, –Zr transforms to –Zr at 1139 K. The – phase boundary for elemental Zr has been studied by high-temperature, static pressure experiments (Jayaraman et al. 1963, Zilbershteyn et al. 1973). The – boundary has been determined upto a pressure of 7.5 GPa (Jayaraman et al. 1963). The –– triple point has been found to occur at 975 K and 6.7 GPa. As mentioned earlier, the -phase appears to be identical to the -phase that occurs at room temperature under high pressures and this implies that the – phase boundary has to turn backward towards the T = 0 K axis at high pressures (Young 1991). Shock wave experiments conducted on Zr are reported to show a discontinuity in the s versus p curve as in the case of Ti and this has been interpreted as being suggestive of the occurrence of a


11

Zr 3

Liquid

T ( × 103 K)

2

bcc ( β)

1

hcp (α)

0

0

hex (ω)

2

4

6

8

10

P (GPa)

Figure 1.2. Experimentally determined temperature–pressure phase diagram for Zr.

phase transition (McQueen et al. 1970, Kutsar et al. 1984). The experimentally determined pressure–temperature phase diagram of Zr is shown in Figure 1.2. In the case of Zr, LMTO calculations predict that the – and – transitions should occur at pressures of 5 GPa and 11 GPa, respectively (Gyanchandani et al. 1990). 1.3.4 Epilogue The occurrence of the -phase at high pressures in elemental Ti and Zr and at room pressures in alloy systems such as Ti–V and Zr–Nb and the similarity of the and the structures have been interpreted as being indicative of the fact that the phase diagrams of Ti and Zr exhibit the phenomenon of s-d electron transfer (Sikka et al. 1982). Effecting an increase in the number of d-electrons, either by the application of pressure or by alloying with elements relatively richer in d-electrons, drives the structure towards the bcc structure characteristic of the next group of elements to the right (i.e. V or Nb). The specific form of the structure, which may be regarded as a hexagonal distortion of the bcc structure, may be related to the details of the Fermi surfaces (Myron et al. 1975, Simmons and Varma 1980). The crystal structures of Ti, Zr and Hf under pressure have recently been studied by Ahuja et al. (1993) by means of first principles, total energy calculations based the local density approximation. These calculations correspond to zero temperature

12


but many of the results obtained by them, especially for Zr, are in good agreement with experimental observations made at room temperature. The observed crystal structure sequence: hcp hP2 → hP3 → bcc cI2 with increasing pressure has been validated for Zr and Hf and it has been predicted that the same sequence should apply to Ti. The equilibrium volumes obtained for Ti, Zr and Hf are 0.0160, 0.0222 and 0 0201 nm3 , respectively, which compare reasonably well with the experimental values of 0.0176, 0.0233 and 0 0223 nm3 for these metals. The calculated c/a values corresponding to the minimum total energy are also in good agreement with the experimental values. Some of the disagreement between the theoretical predictions and the room temperature experimental observations could be ascribed to thermal effects. For example, the calculations indicate that at the theoretical equilibrium volume, the hP3 structure is slightly more stable than the hP2 structure; but room temperature observations show that the reverse is true—a result that matches with the calculations at the experimental volume. An important point is that the calculations do show that the energy difference between the - and the -phases is small for both Ti and Zr, which is consistent with the fact that the pressure induced → transition can be brought about in these metals at moderately high pressures. The calculations of Ahuja et al. (1993) indicate that the charge density for the -phase has a substantial non-spherical component, reflecting covalent bonding. This is quite different from the chemical bonding prevailing in the fcc, hcp and bcc structures where the charge density is predominantly spherical around the atomic positions and flat in the intervening regions. The chemical bonding for these structures is metallic. However, despite the difference in the nature of the chemical bonds for the various structures, band filling arguments can be used, at least to some extent, to explain the crystallographic sequence encountered in these tetravalent metals. At zero temperature and sufficiently high pressures, all the three metals – Ti, Zr and Hf – are predicted to assume the bcc structure. Again, at zero pressure and high temperaturess, these elements are known to transform from the hcp to the bcc structure. There is thus the possibility that the two bcc regions in a pressure–temperature phase diagram will be in contact. A schematic phase diagram, pertinent to these metals, has been constructed by Ahuja et al. (1993) and is shown in Figure 1.3. These authors have also examined the issue of the stability of the bcc phase. They have shown that the tetragonal shear constant, C = C11 − C12 /2, has a negative value at zero pressure for the bcc structure. This corresponds to a mechanically unstable situation. However, the sign of C changes with increasing pressure. For the high pressure bcc phase, the calculated C values are all positive, in agreement with the observed high pressure bcc phase in Zr and Hf. This can be explained as an effect of s–d electron transfer; for example, the d-band occupation


13

Temperature

L

bcc

I

II

hcp

ω

bcc

Pressure

Figure 1.3. Schematic temperature–pressure phase diagram for the metals Ti, Zr and Hf. The bcc phase is mechanically unstable in the region I and mechanically stable in the region II at low temperatures.

of Zr increases under pressure, making it behave more like the element to its right, i.e. Nb, which has a bcc crystal structure. Even though the bcc structure, according to calculations, is mechanically unstable at zero pressure, the high temperature -phase of all the three metals is known to posses this structure. This can be explained in terms of the high entropy associated with the bcc structure. The -phase of these elements shows some anomalous properties including its well known anomalously fast diffusion behaviour. This behaviour might be related to the intrinsic mechanical instability associated with the value of the C parameter. Another possible explanation suggested for the anomalous diffusion behaviour invokes -embryos acting as activated complex configurations in the atom–vacancy exchange process (Sanchez and de Fotnaine 1975). The fact that the -phase is calculated to have a lower total energy than the -phase at the equilibrium volume for all the three metals lends support to such an interpretation. The mechanical instability of the bcc phase becomes less severe with increasing pressure in the sense that the value of C becomes less negative with decreasing volume. Therefore, as the pressure increases, a progressively lower temperature is needed to restore the stability of the bcc structure (Ahuja et al. 1993). 1.3.5 Phase stability and electronic structure The stability of phases, the dependence of this stability on parameters like temperature and pressure and the selection of phases that are actually observed and recorded in phase diagrams are determined by the result of the competition among several possible phases (and, therefore, structures) that could be stable in a given

14


system. This competition is based on the respective values of the Gibbs free energy corresponding to the various pertinent phases and their variation with temperature, pressure, composition and parameters such as magnetic, electric or stress fields, dose rates of particle and photon irradiation, etc. A number of factors contribute to the enthalpy, H, and the entropy, S. A very important contribution to the entropy arises from the statistical mixing of atoms. There may be additional contributions from vibrational effects, clustering of atoms, distribution of magnetic moments, long range configurational effects, etc. The statistical mixing of atoms contributes to the enthalpy as well. These contributions are related to the interaction energies: those corresponding to nearest neighbour atoms, next nearest neighbour atoms and further distant atoms in a given structure. These interaction energies may arise from various origins – electronic, magnetic, elastic and vibrational. A formidable problem in the context of the assessment of phase stability is that the relative stability among the competing crystal structures is usually dictated by very small energy differences between large values of the cohesive energy. Apart from this, a correct prediction implies the prediction of the lowest free energy structure among the chosen structural alternatives. This, in turn, stipulates a prior algorithm to generate all probable structures. Even when all these difficulties are overcome, it is needed to incorporate the roles of variables like temperature and pressure in realistic terms. These are, indeed, difficult tasks. The success of a theory of phase stability is largely determined by its ability to make predictions that are consistent with experimental observations. There is a need to be able to calculate phase stability from “first principles” if the basic microscopic parameters that dictate the free energy of a phase are to be properly understood. It should also be possible to make use of such calculations for predicting phase diagrams in systems where the experimental determination of such diagrams is difficult. The understanding and prediction of phase stability in respect of disordered and ordered alloys in terms of electronic structure calculations constitute an area of considerable importance in materials science and significant progress has been made with regard to the “first principles” approach to the band theory of such materials (Massalski 1996). The computation of an alloy phase diagram from first principles implies its delineation from a knowledge of the electronic structure of the alloy. In a truly ab initio calculation, one begins with a periodic array of nuclei of charge Ze together with Z electrons per nucleus, and then solves the Schrodinger equation for the total energy of the system. When Z is small (e.g. for H, He and Li), it is possible to handle this problem by Quantum Monte Carlo (QMC) methods (Ceperley and Alder 1986) which are exact in principle. However, the QMC method is not yet practical for heavier atoms, and the development of the density functional theory and its computational version, the local density approximation (LDA), has been of


15

great value. Here the full many-body wave function is approximated as a product of one-electron functions, and the exchange–correlation energy is expressed as a function of the local electron density, nr, given by nr = k r2 k r being the one-electron wave function for the occupied state k (Young 1991). In the density functional theory, the total energy of a system of nuclei and electrons is considered to be a unique functional of nr and is a minimum at the true ground state. The total energy, Et , is expressed as Et = E1 + E2 + E3 + E4 + E5 where the terms on the right-hand side represent the kinetic (E1 ), electron–nucleus (E2 ), electron–electron (E3 ), exchange–correlation (E4 ) and nucleus–nucleus (E5 ) energies. The different approaches used to solve the one-electron Schrodinger equation, with the imposition of the lattice periodicity (Bloch condition) as a boundary condition, have engendered a variety of band-structure methods; some of these are (Young 1991): ab initio pseudopotential (AIP); linear muffin tin orbital (LMTO); augmented spherical wave (ASW); linearized augmented plane wave (LAPW); full-potential LAPW (FPLAPW) and linear combination of Gaussian-type orbitals (LCGTO). The LMTO method, which has been extensively used, is based on some additional approximations. While the muffin tin (MT) potential implies that the atomic potential Vr is spherically symmetric within a sphere inscribed in the primitive unit cell and is constant in the interstitial region, the LMTO method brings in a further simplification by way of the atomic-sphere approximation (ASA), whereby the spherical potential is extended to the full atomic volumes, reducing the net interstitial volume to zero. The Bloch condition is implemented by effecting the cancellation of all neighbour wave functions within the atomic sphere (Skriver 1984). The ‘L’ in LMTO implies the approximation that the basis functions are made energy-independent; this permits the eigenfunctions to be obtained in a single diagonalization operation, speeding up the calculation enormously and thus contributing to the efficacy of the method, a major limitation of which is the restriction to high-symmetry crystal structures imposed by the ASA (Young 1991). The LMTO method has been used to predict the stability of different phases with regard to the pressure–temperature phase diagrams of many transition metals, including Ti and Zr. Total energy calculations based on the LDA, which use only atomic numbers as inputs, have been very successful in the estimation of 0 K ground state properties of the elements and of ordered compounds. In fact, the implementation of the LDA by many an investigator, combined with the development of efficient linear methods for studying the electronic structure of solids, has led to fully ab initio calculations of the total energy at 0 K of pure solids, relatively simple compounds and disordered alloys (Sanchez 1992). By making it possible to assess a wide range

16


of physical properties quite close to the corresponding experimentally obtained values, these quantum mechanical total energy computations have provided very favourable evidence in support of the LDA method, which can be applied, together with appropriate statistical models, to address the difficult problem of alloy stability at non-zero temperatures. Even though the LDA method has been quite successful, it has some non-trivial limitations including the underestimation of band gap energies and the inability to predict narrow band Mott-transition phenomena (Young 1991). A more general method for calculating the equilibrium state of matter at finite temperatures is the quantum molecular dynamics method (Car and Parrinello 1989). In this approach, the LDA wave function is solved for a small number of nuclei in an arbitrary configuration and the Hellman–Feynman theorem is used for finding the net force on each nucleus; the nuclei are then moved in accordance with classical (Newtonian) dynamics and the LDA calculation is undertaken again for the new configuration of the nuclei. This approach has been found to be useful for arriving at band structures and bonding details in respect of solids and liquids at finite temperatures (Young 1991). In the context of statistical models, it is appropriate to make a mention here of the Monte Carlo (MC) (Binder 1986) and molecular dynamics (MD) (Hoover 1986) methods. Like QMC, these methods are exact in principle. Although it is possible to undertake direct calculation of free energy by MC, the technique is not yet very suitable for the determination of phase stability and accurate delineation of phase boundaries. As of now, the MD method also suffers from similar limitations. It is true that isobaric–isothermal ensemble versions of MC and MD have been successfully employed to predict the most stable crystal structures of certain solids (Parrinello and Rahman 1981), but these methods have found their most important use in providing a standard for comparing and refining approximate statistical mechanics models (Young 1991). Some of the aspects briefly outlined in the preceding paragraphs have been covered in greater detail in a subsequent chapter. It is to be noted that a major shortcoming of many of the ab initio phase diagram calculations concerns the inadequate treatment of local volume and elastic relaxations and the neglect of vibrational modes. Even in crystalline solids, atoms are in perpetual motion; they move from one lattice site to another by diffusion at non-zero temperatures and also vibrate about their equilibrium positions. In a multicomponent system like an alloy, a given lattice site is occupied by atoms of different species at different times. If a large atom replaces a small one, the environment of the lattice site responds by expanding. Likewise, when a small atom replaces a large atom, the neighbouring atoms relax towards the lattice site in question. It should be possible to address the accompanying strain fluctuations


17

within the same type of first principles framework that is pertinent to fluctuations in concentration. However, the treatment of local relaxations of this kind presents a very difficult problem and not many attempts appear to have been made to include this effect in first principles calculations of phase stability and phase diagrams (Sanchez 1992, Gyorffy et al. 1992). Apart from the direct quantum mechanical route, many semi-empirical schemes pertaining to phase stability have also been pursued, often with a good deal of success. Many of these schemes involve the construction of certain phenomenological scales on which various aspects of bonding and structural characteristics are measured (Raju et al. 1995). These scales include parameters like the electronegativity factor, the size factor, the coordination factor, the electron concentration (e/a) factor, the promotion energy factor, etc. that are used to systematize a variety of structural features. The resulting structure maps are essentially graphical representations of the relative structural stability of alloy phases. They are two-dimensional diagrams, constructed by using suitable alloy theory coordinates for sorting out different crystal structures that are compatible with a chosen alloy stoichiometry. The efficacy of these structure maps depends crucially on the appropriate choice of coordinates. What are needed are those “bond indicators” which are transparent in their physical content, are transferable in their applicability and have a bearing on the alloy formation situation in terms of a validated model (Raju et al. 1995). In the classical approach, the emphasis has been on the construction of physically simple and transferable coordinates that may systematize the observed trends in relation to the occurrence of alloy phases. The major limitations of the classical formalism lie in the linear dependencies among many of the different phenomenological scales and the absence of a microscopic model that connects one or more of these directly to a real space alloy physics (Raju et al. 1995). Quantum mechanical considerations have been invoked in order to tide over these deficiencies with the result that the classical coordinates have been replaced by what are known as quantum structural parameters and classical structure diagrams by quantum structural diagrams (QSD). There have been numerous applications of QSD to various classes of solids including intermetallics, quasicrystals, high Tc superconductors and permanent magnetic materials (Phillips 1991). Even though not all of these have served to elucidate the issue of structural stability of condensed phases, these have been very useful in ordering the vast available data base into certain systematics. There are, indeed, quite a few examples of QSD which have really enhanced the understanding of the physicochemical factors governing phase stability. Most of the existing models pertaining to phase stability, ranging from those offering detailed density maps and electronic parameters of alloys to the semiempirical ones, suffer from a major difficulty in the context of the construction

18


of phase diagrams in that a theoretical treatment of the temperature dependence of energy is not straightforward and tractable (Massalski 1996). The calculations used for predicting enthalpy at 0 K (first principles calculations) or at some undefined temperature (semi-empirical models) are seldom able to furnish adequate information regarding the thermal behaviour of such enthalpies or the thermal entropy contributions to the free energy. The prediction of entropies, particularly for relevant metastable phases in phase diagrams, has to be realized for the utilization of the full potential of the theoretical methods of phase stability calculations. 1.3.6 Some features of transition metals Elements belonging to the family of transition metals, of which Ti and Zr are members, are generally characterized by certain interesting features. Some of these will be briefly covered in this section. Elements of Groups 3–10 in the periodic table constitute the transition metals which have in common that their d-orbitals (3d, 4d and 5d) are partially occupied. These orbitals are only slightly screened by the outer s-electrons, resulting in significantly different chemical properties of these elements going from left to right in the periodic table; the atomic volumes rapidly decrease with increasing number of electrons in the bonding d-orbitals, because of cohesion, and then increase as the anti-bonding d-orbitals get filled (Steurer 1996). Transition metals are characterized by a fairly tightly bound (and partially filled) d-band that overlaps and hybridizes with a broader nearly-free-electron (NFE) sp-band. The d-band (with a large density of states near the Fermi level) is well described within the tight-binding (TB) approximation by a linear combination of atomic d-orbitals and the difference in behaviour between the valence sp and d electrons arises from the d-shell lying inside the outer valence s-shell, thereby resulting in a small overlap between the d-orbitals in the bulk (Pettifor 1996). In general, the transition metals exhibit high densities, cohesive energies and bulk moduli, with some exceptions. These characteristics arise from strong d- electron bonding. Plots of molar volume, cohesive energy and bulk modulus against the number of d-electrons yield roughly symmetrical curves with extreme values approximately at the middle of the series (Young 1991). An exception to this trend occurs with the 3d magnetic elements. The values of these parameters for the transition elements are shown in Table 1.5. The general behaviour alluded to the above can be rationalized in terms of the Friedel model of transition metal d-bands (Harrison 1980). Cohesive energy versus group number plots for 3d, 4d and 5d transition metals are shown in Figure 1.4. The sequence of the observed room temperature (and pressure) crystal structures in the case of 3d, 4d and 5d transition metals is presented in Table 1.6. This


19

Table 1.5. Values of molar volume, cohesive energy and bulk modulus for transition metals (Young 1991). Z

Element

21 22 23 24 25 26 27 28 39 40 41 42 43 44 45 46 71 72 73 74 75 76 77 78

Sc Ti V Cr Mn Fe Co Ni Y Zr Nb Mo Tc Ru Rh Pd Lu Hf Ta W Re Os Ir Pt

Molar volume (m3 /M mol)

Cohesive energy (kJ/mol)

Bulk modulus (GPa)

15 00 10 64 8 32 7 23 7 35 7 09 6 67 6 59 19 88 14 02 10 83 9 38 8 63 8 17 8 28 8 56 17 78 13 44 10 85 9 47 8 86 8 42 8 52 9 09

376 0 467 0 511 0 395 0 282 0 413 0 427 0 428 0 424 0 607 0 718 0 656 0 688 0 650 0 552 0 376 0 428 0 619 0 781 0 848 0 774 0 788 0 668 0 564 0

54 6 106 0 155 0 160 0 90 4 163 0 186 0 179 0 41 0 94 9 169 0 261 0 − 303 0 282 0 189 0 47 4 108 0 191 0 308 0 360 0 − 358 0 277 0

observed sequence (hcp → bcc → hcp → fcc) indicates that close packed structures are preferred at either end of the series, while the more open bcc structure is preferred in the middle. Pettifor (1977) has carried out a TB orbital calculation and shown that the structure sequence across the series is the result of the filling of the d-band and that the s-p electron number is nearly constant. While this model correctly predicts the structure sequence hcp → bcc → hcp → fcc, it does not predict the structures of all the elements correctly. In the tight binding model, to a first-order approximation, the cohesive energy turns out to be independent of structure; the relative structural stability arises from small differences in band structure contribution to the total electronic energy, an adequate description of which calls for the inclusion of higher order moments for describing the density of states curve (Raju et al. 1996). A fully self consistent LMTO calculation leads to a still better agreement between theory and experiment (Skriver 1984).

20

Phase Transformations: Titanium and Zirconium Alloys 950

Cohesive energy (kJ/mol)

850 750

5d

650

4d 550 450

3d 350 250

3

4

5

6

7

8

9

10

Group number

Figure 1.4. Cohesive energy versus group number plots for 3d, 4d and 5d transition metals. Table 1.6. Crystal structures of d-transition metals at room temperature and pressure.

3d series 4d series 5d series

HCP

BCC

HCP

FCC

Sc Ti Y Zr Lu Hf

V Cr Fe Mn Nb Mo Ta W

Co Tc Ru Re Os

Ni Rh Pd Ir Pt

Note: The actual structure of Mn is complex though it is listed under bcc in this table.

A systematic theoretical study with regard to the phase transitions that can be expected to occur in unalloyed transition metals at ultra-high pressures has not yet been attempted. However, it is, in general, expected that the early transition metals will assume the structures of their right-hand side neighbours as the s–d electron transfer will lead to the filling of the d-band under pressure; for the later members of the series, pressure is expected to have the effect of emptying the d-band, thus reversing the earlier trend (Young 1991). Obviously, transition metal phase transitions can also be driven by alloying, whereby the number of electrons populating the d-band can be altered. Fairly general theoretical arguments suggest that alloys of transition metals with roughly half-filled d-bands exhibit ordering tendencies, while those with nearly empty or nearly full d-bands show clustering tendencies in the disordered state and thus tend to phase separate at low temperatures; this prediction appears to be borne out by a considerable body of experimental data, even though there are many exceptions to this rule (Gyorffy et al. 1992).


1.4

21

EFFECT OF ALLOYING

1.4.1 Introductory remarks In alloys based on Ti or Zr, a very important effect of an alloying element pertains to the manner in which its addition affects the allotropic -phase to -phase transformation temperature. Some elements stabilize the -phase by raising this temperature while some others lower it, thereby stabilizing the -phase. Elements which, on being dissolved in Ti or Zr, cause the transformation temperature to increase or bring about little change in it are known as -stabilizers. These elements are generally non-transition metals or interstitial elements (like C, N and O). Elements which, on alloying with Ti or Zr, bring down the transformation temperature are termed -stabilizers. These elements are generally the transition metals and the noble metals with unfilled or just filled d-electron bands. Among the interstitial elements, H is a -stabilizer. Unlike in pure Ti or Zr, in alloys the single phase and the single phase regions are separated by a two-phase + region in the temperature versus composition phase diagram. The width of this region increases with increasing solute content. The single equilibrium - to -phase transformation temperature associated with elemental Ti or Zr is replaced by two equilibrium temperatures in the case of an alloy: the -transus temperature, below which the alloy contains only the -phase, and the -transus temperature, above which the alloy contains only the -phase. At temperatures between these two temperatures, both the - and the -phases are present. 1.4.2 Alloy classification The allotropic transformation exhibited by Ti and Zr forms the basis of the classification of commercial alloys based on these metals. Such classification is effected on the basis of the phases present in these alloys at ambient temperature (and pressure). The relative proportions of the constituent phases are determined by the nature (-stabilizing or -stabilizing) and the amounts of the alloying elements. In the case of alloys, the - and -phases contain various amounts of the different alloying species in solid solution. 1.4.3 Titanium alloys Technical alloys of Ti, which are generally multicomponent alloys containing -stabilizing as well as -stabilizing elements, are broadly classified as alloys, + alloys and alloys. Within the second category, there are the subclasses “near ” and “near ” alloys, referring to alloys whose compositions place them near the / + or the + / phase boundaries, respectively.

22


Unalloyed Ti and its alloys with one or more -stabilizing elements consist fully or predominantly of the -phase at room temperature and are known as alloys. The -phase continues to be the primary phase constituent of most of these alloys at temperatures well beyond about 1040 K (Froes et al. 1996). These alloys generally exhibit good strength, toughness, creep resistance and weldability, together with the absence of a ductile-to-brittle transition (Collings 1984). However, they are not amenable to strengthening by heat treatment. The compositions of + alloys are such that at room temperature they contain a mixture of the - and -phases. These alloys have one or more of - as well as -stabilizing elements as alloying additions. In general, + alloys possess good fabricability. They are very strong at room temperature and moderately so at high temperatures (Collings 1984). The relative volume fractions of the - and -phases in these alloys can be varied by heat treatment, which provides a handle for adjusting their properties. In -alloys, the -phase is stabilized by the addition of adequate amounts of -stabilizing elements and can be retained at room temperature. These alloys generally contain significant amounts of one or more of the transition metals V, Nb, Ta (Group 5) and Mo (Group 6). These “-isomorphous” alloying elements do not form intermetallic compounds through eutectoid decomposition of the -phase and are generally preferred to eutectoid forming -stabilizing elements such as Cr, Cu, Ni; however, elements of the latter category are sometimes added to (and + ) alloys for improving their hardenability and response to heat treatment (Froes et al. 1996). The strength of alloys is generally greater than that of + and -alloys. Moreover, they exhibit excellent formability (Wood 1972). But they have relatively high densities, are prone to ductile–brittle transition at low temperatures and generally possess inferior creep resistance as compared to and + alloys (Collings 1984, Froes et al. 1996). The archetypical -stabilizing and -stabilizing alloying additions to Ti are Al and Mo, respectively. It is useful to be able to describe a multicomponent Ti-based alloy in terms of its “equivalent” Al and Mo contents. The two pertinent expressions often quoted in this context (Collings 1994) are:

Aleq = Al + Zr/3 + Sn/3 + 10 O

Moeq = Mo + Ta/5 + Nb/3 6 + W/2 5 + + V/1 25 + 1 25 Cr + 1 25 Ni + 1 7 Mn + 1 7 Co + 2 5 Fe where [X] indicates the concentration of the element X in weight per cent in the alloy. It can be seen that while Al and O are strong -stabilizers, Sn and Zr are relatively weak ones. It can also be seen that the efficacy of the transition elements


23

with regard to the stabilization of the -phase progressively increases in the order: Ta, Nb, W, V, Mo, Cr and Ni, Mn and Co, and Fe, the last being the strongest -stabilizer. It may be mentioned here that Ti can form extensive substitutional solid solutions with most of the elements with atomic size factor within about 20% and this fact has opened up many alloying possibilities for exploitation. Some examples of important commercial Ti base alloys are: Ti-5Al-2.5Sn ( alloys); Ti-8Al-1Mo-1V, Ti-6Al-2Sn-4Zr-2Mo (near alloys); Ti-6Al-4V, Ti-6Al-2Sn-6V, Ti-3Al-2.5V ( + alloys); Ti-6Al-2Sn-4Zr-6Mo, Ti-5Al-2Sn2Zr-4Cr-4Mo, Ti-3Al-10V-2Fe (near alloys); Ti-13V-11Cr-3Al, Ti-15V-3Cr3Al-3Sn, Ti-4Mo-8V-6Cr-4Zr-3Al, Ti-11.5Mo-6Zr-4.5Sn ( alloys). 1.4.4 Zirconium alloys Unlike Ti, Zr is not quite amenable to alloying. One of the reasons for this could be the relatively large size of the Zr atom. Most of the elements have very limited solubilities in -Zr, with a few exceptions such as Ti, Hf, Sc and O. By comparison -Zr is a much better solvent, but it is generally quite difficult to retain the -phase at room temperature in a metastable state by quenching (Froes et al. 1996). The occurrence of non-equilibrium phases in -quenched Ti- and Zr-based alloys has been dealt with in a later section. According to the exhaustive compilation made by Douglass (1971), the retention of the -phase during quenching has been found to be feasible in the binary Zr–Mo, Zr–Cr, Zr–Nb, Zr–U, Zr–V and Zr–Re systems. The minimum concentrations of alloying additions for complete retention of the -phase in the first four systems are 5 wt%, 7.2 wt%, 15 wt% and 20 wt% respectively. Retention of cent per cent -phase is not possible in the systems Zr–V and Zr–Re; alloys containing the maximum amounts of V or Re in solution at quenching temperatures as high as 1573 K have been found to contain the -phase in addition to the -phase (Petrova 1962). The retention of quite large volume fractions of a metastable, Zr-rich 1 -phase has been observed in relatively solute-lean alloys (Zr-2.5 wt% Nb and Zr-5 wt% Ta) belonging to the monotectoid Zr–Nb (Banerjee et al. 1976, Menon et al. 1978) and Zr–Ta (Mukhopadhyay et al. 1978, Menon et al. 1979) systems. The most common Zr alloys of commercial importance are the zircaloys, namely zircaloy 2: Zr-1.5Sn-0.1Cr-0.1Fe-0.1Ni, Cr + Fe + Ni not to exceed 0.38 wt%; zircaloy 4: Zr-1.5Sn-0.15Cr- 0.15Fe, Cr + Fe not to exceed 0.3 wt% and the Zr-2.5% Nb, Zr-1% Nb and Zr-2.5Nb-0.5Cu alloys. These alloys contain only small amounts of -stabilizing elements and are all basically -alloys, with the -phase as the predominant constituent phase.

24


1.4.5 Stability of titanium and zirconium alloys The aspect of lattice stability or, in other words, of structural phase stability is an important issue with regard to pure metals like Ti and Zr and alloys based on these. It has been stated in an earlier section that the crystal structures of the three long periods of transition metals follow the sequence hcp → bcc → hcp → fcc as the group number increases from 3 to 10 (3d: Sc to Ni; 4d: Y to Pd; 5d: Lu to Pt). It appears that there is a correlation between the crystal structure and the group number in the case of the elemental transition metals and between the crystal structure and the average group number or the electron to atom (e/a) ratio in the case of alloys. The occurrence of correlations like this testifies to the fact that the electronic structure is a key factor in determining phase stability. The e/a ratio is a parameter which relates to many properties of binary transition metal alloys, particularly Ti–X alloys, where X represents a transition metal (Collings 1984). A qualitatively similar situation is obtained with Zr–X alloys also. However, a general and comprehensive theoretical explanation rationalizing the correlation between phase stability and electron concentration (which is the same as or is closely related to the e/a ratio) in the case of transition metal systems is still to evolve (Faulkner 1982). The issue of the stability of equilibrium phases in Ti (and Zr) alloys can also be addressed by adopting a thermodynamic approach (Kaufman and Bernstein 1970, Kaufman and Nesor 1973). In this approach, the energywise competition between the relevant phases is duly considered while assessing phase stability in unalloyed metals as well as in alloys. This quantitative thermodynamic approach has been used for the computation of phase diagrams pertaining to binary as well as multicomponent systems. It has been mentioned earlier in the context of Ti–X and Zr–X alloys that -phase stabilizers are generally non-transition or simple metals, while -phase stabilizers are generally transition metals and noble metals. Collings (1984) has put forward a qualitative explanation, based on electron screening considerations, with regard to the phase stabilizing action of -stabilizer and -stabilizer solutes. This is outlined in the following paragraphs. When a simple metal X is dissolved in Ti (or Zr), most of the electrons belonging to X atoms occupy states in the lower part of the band and only very few appear at the Fermi level. The d-electrons belonging to the host (solvent) tend to avoid the solute atoms and this leads to a dilution of the Ti (or Zr) sublattice. A consequence of this is to emphasise any pre-existing Ti–Ti (or Zr–Zr) bond directionality and thereby to preserve the hcp structure characteristic of Ti (or Zr). As more and more X atoms are added, the field of Ti- (or Zr)-like -stability is ultimately terminated, generally by the appearance of an intermetallic phase of the stoichiometry Ti3 X (or Zr 3 X), which is also based on or is closely related to the hcp structure.


25

Coming next to the case of -phase stabilization, one may first recall that the crystal structures of transition metals change from hcp to bcc as the e/a ratio increases from 4 to 6. Collings (1984) has pointed out that it is possible to rationalize this stabilization of the bcc structure within the framework of an electron screening model which stipulates that a high concentration of conduction electrons, by enhancing the screening of ion cores, may cause a symmetrical (i.e. cubic) crystal structure to be favoured. Thus an increase in the electron density (as in elements belonging to Groups 5 and 6) will tend to symmetrize the screening, thereby enhancing the stability of the bcc structure. The fact that the six d-transition metals belonging to Groups 3 and 4 undergo the hcp → bcc structural transformation at high temperatures indicates that symmetrization can also be accomplished through lattice vibrations (Collings 1984). Given this background, one can see that the addition of transition metals belonging to Groups 5–10 to Ti or Zr increases the electron density and as a consequence, stabilizes the bcc or -phase. Thus, such elements are -stabilizers. Ageev and Petrova (1970) have pointed out in the context of Ti alloys that the -stabilization brought about by transition metal solutes is more effective the farther they are from Ti in the periodic table and that for the retention of the -phase during quenching from the -phase field, the nature and the concentration of the -stabilizer has to be such that the value of the e/a ratio is at least 4.2. In the context of the stability of bcc transition metals, it has been shown (Fisher and Dever 1970, Fisher 1975) that the magnitude of the elastic shear modulus C , defined as C11 − C12 /2, can be used for comparing the stabilities of these metals and their alloys. A cubic monocrystal is characterized by three fundamental stiffness moduli, C11 C44 and C12 . The shear stiffness modulus, C , though made up of two fundamental moduli, is obtainable directly by experiment. The ultrasonic waves needed for the measurement of these moduli are (Collings 1984): a longitudinal wave in a direction for C11 ; a transverse wave in a direction, polarized along < 100 > or a transverse wave in a < 100 > direction, polarized along < 100 > for C44 ; and the other transverse wave in a < 100 > direction, ¯ > for C . Since C44 is governed by the transverse polarized along < 110 ¯ wave, polarized, and C by the same wave, < 110> polarised, C = C44 in an isotropic cubic material. Collings and Gegel (1973) have studied the variation of the parameter C with the e/a ratio and have demonstrated that alloying Group 4 elements with elements occurring to the right of them in the periodic table enhances the stability of the bcc structure and that this effect is maximized at about e/a = 6 (for the elements Cr, Mo and W). They have also found that C almost vanishes at e/a = 4 1 and that this value corresponds to the compositional threshold for martensitic transformation. In an anisotropic cubic material, the extent of the departure from isotropy is indicated by the value of the so called Zener anisotropy

26


ratio, A = C44 /C . While in simple bcc metals like Na, the values of A are quite large, these can be quite low for bcc transition metals; for example, for the Group 6 metals Cr, Mo and W, the values of A are 0.71, 0.72 and 1.01, respectively (Fisher 1975). Fisher (1975) has also pointed out that while the C44 shears are resisted primarily by nearest neighbour repulsion, the C shear depends mainly on the next nearest neighbour forces. The large values of C for bcc transition elements are thought to be a consequence of the cohesive contributions of the d-electrons. The parameter C appears to be interpretable as a bcc stability parameter. Thus, for the highly stable bcc transition metals of Group 6, C is about 1 5 × 1011 N/m2 but its values decrease rapidly with decreasing e/a ratio, approaching zero at room temperature for alloys which exhibit -phase instabilities or under a martensitic transformation at ordinary temperatures (Collings and Gegel 1973). When Ti (or Zr) is alloyed with transition metals of higher group numbers, the increasing stability of the -phase is reflected in a continuous lowering of the / + transus temperature. It is mentioned later in this chapter that in the case of -stabilized binary Ti alloys, two types of phase diagrams are encountered: -isomorphous and -eutectoid. Collings (1984) has pointed out that a general trend is that as the group number of the solute increases, there is a tendency for the phase diagram to change from the former to the latter type.

1.5

BINARY PHASE DIAGRAMS

1.5.1 Introductory remarks Binary Ti–X and Zr–X (X being any element other than Ti and Zr, respectively) phase diagrams exhibit multifarious forms and reflect various kinds of phase reactions. The equilibrium phases are the - and -phases and numerous intermetallic phases. These are the phases that are shown in the equilibrium phase diagrams. However, many non-equilibrium phases such as the martensite phase (hcp and orthorhombic), the -phase and a large number of metastable intermetallic phases also occur in binary Ti and Zr base alloys. Some of these will be covered in detail in the succeeding chapters. There have been many attempts to categorize Ti and Zr alloy phase diagrams, taking cognizance of the fact that basically there are two types of systems, namely -stabilized and -stabilized systems. As mentioned earlier, in the former case X is usually a non-transition or simple metal, while in the latter X is usually a transition or a noble metal. It has been suggested in the context of Ti–X systems that the regular solution thermodynamic interaction parameter, ij , is positive for -stabilized alloys, indicating a clustering tendency, and negative for -stabilized alloys, indicating an ordering tendency (Collings and Gegel 1975).


27

For a given element X, the differences in the nature of the binary Ti–X and Zr–X equilibrium phase diagrams generally arise from the relative inefficiency of -Zr and -Zr with regard to taking X in solid solution as compared to -Ti and -Ti, particularly when X is a substitutional element. 1.5.2 Ti–X systems Margolin and Nielsen (1960) have suggested that -stabilized Ti–X systems can be basically subdivided into three classes: (a) –-isomorphous systems where X is completely soluble in the - as well as -phases (e.g. Ti–Zr, Ti–Hf); (b) -isomorphous systems where X is completely soluble in the -phase and has limited solubility in the -phase (e.g. Ti–V, Ti–Mo) and (c) -eutectoid systems where X has a limited solubility in the -phase which decomposes eutectoidally into the -phase and an appropriate intermetallic phase, Tim Xn , on cooling. Depending on the kinetics of -phase decomposition, this class is further subdivisible into “active” (rapid, e.g. Ti–Cu, Ti–Ni) and “sluggish” (e.g. Ti–Cr, Ti–Mn) eutectoid systems. They have also suggested that -stabilized Ti–X systems can be subdivided into two categories, depending on the degree of -phase stabilization: (a) systems exhibiting a “limited” degree of -stability, where the -phase is related to the - and an appropriate intermetallic phase by a peritectoid reaction (e.g. Ti–B, Ti–Al); and (b) systems characterized by a “complete -phase stability” where the -phase can coexist with the liquid phase (e.g. Ti–N, Ti–O). An exhaustive classification scheme for binary Ti–X phase diagrams has subsequently been suggested by Molchanova (1965) who has classified the available equilibrium phase diagrams into three broad groups, each of which contains a few subgroups. This classification, as reported by Collings (1984), is shown below: Group I: Systems where X shows continuous solid solubility in the -phase Subgroup I (a): Complete solubility in the -phase (X: Zr, Hf) Subgroup I (b): Partial solubility in the -phase (X: V, Nb, Ta, Mo) Subgroup I (c): Partial solubility in the -phase and eutectoid decomposition of the -phase (X: Cr, U) Group II: Eutectic systems Subgroup II (a): Partial solid solubility in the - and -phases; eutectoid decomposition of the -phase (X: H, Cu, Ag, Au, Be, Si, Sn, Bi, Mn, Fe, Co, Ni, Pd, Pt) Subgroup II (b): Partial solid solubility in the - and -phases; peritectoid – transformation (X: B, Sc, Ga, La, Ce, Nd, Gd, Ge) Subgroup II (c): Extremely limited solid solubility in the - and -phases (X: Y, Th)

28


Group III: Peritectic systems Subgroup III (a): Simple peritectic (X: N, O) Subgroup III (b): Partial solid solubility in the - and -phases (X: Re) Subgroup III (c): Partial solid solubility in the - and -phases; eutectoid decomposition of the -phase (X: Pb, W) Subgroup III (d): Partial solid solubility in the - and -phases; peritectoid – transformation (X: Al, C). In a simpler classification, Molchanova (1965) has suggested that binary Ti–X equilibrium phase diagrams can be divided into four categories: -isomorphous (including – isomorphous), comprising subgroups I (a), I (b) and III (b); -eutectoid, comprising subgroups I (c), II (a) and III (c); simple peritectic, comprising subgroup III (a); and -peritectoid, comprising subgroups II (b) and III (d). This classification scheme is shown in Figure 1.5 in which the legends , and stand for the -phase, the -phase and the pertinent intermetallic phase, respectively.

Binary Ti alloys

β-stabilized

α-stabilized

β-isomorphous

β-eutectoid

Simple peritectic

β-peritectoid

Solutes V,Zr,Nb,Mo, Hf,Ta,Re

Solutes Cr,Mn,Fe,Co,Ni,Cu Pd,Ag,W,Pt,Au H,Be,Si,Sn,Pb,Bi,U

Solutes N,O

Solutes B,Sc,Ga,La Ca,Gd,Nd,Ge Al,C

Ti

L+α

L

β

β

α

γ

L+

β

L+

L+β

L

β+γ

β+α

β

α

β+γ

α+

Ti

β

β

α

L L+γ β

α

β

α+

Temperature

L+

β+

L+β

L

α+γ

α+γ Ti Solute content

α

α+γ

Ti

Figure 1.5. A classification scheme for binary Ti–X equilibrium phase diagrams. The legends and stand, respectively, for the -phase, the -phase and the pertinent intermetallic phase.


29

It is to be noted that quite a few Ti–X systems, designated earlier as -isomorphous systems, are not so in reality (Massalski et al. 1992). Below a transus delineating the upper boundary of a region referred to as a “miscibility gap”, a homogeneous, single-phase -solid solution decomposes into a thermodynamically stable aggregate of two bcc phases, one Ti-rich (1 ) and the other solute-rich 2 → 1 + 2 . The former participates in a monotectoid reaction: 1 → + 2 , the monotectoid temperature and composition varying from system to system. Examples of Ti–X systems where such a monotectoid reaction occurs include Ti–V, Ti–Mo, Ti–Nb and Ti–W. 1.5.3 Zr–X systems It has been pointed out earlier that inspite of the similarity in the electronic and crystal structures of Ti and Zr (both of these transition metals belong to Group 4 of the periodic table of elements), the alloying behaviour of these elements exhibit noteworthy differences, largely due to the size factor. While one encounters the – isomorphous, -eutectoid and -stabilized types of equilibrium diagrams in Zr–X systems, -isomorphous type phase diagrams do not occur in these alloys. Alloying elements, X, which give rise to -isomorphous equilibrium phase diagrams with Ti, yield either -eutectoid (e.g. X: V, Mo, Re) or -monotectoid (e.g. X: Nb, Ta) types of equilibrium diagrams with Zr. For a Pauling valence of 4, the second Brillouin zone is the one most nearly filled for in the case of Zr. This zone for -Zr is bounded boththe -and -phases ¯ and 1120 ¯ planes and has a volume of 3.6 electrons per atom; by the 1012 ¯ side the excess electrons, 0.4 per atom, overlap into the third zone on the 1012 of the second zone (Luke et al. 1965). The second Brillouin zone for the -phase is bounded by 200 and 211 planes and has a volume of eight electrons per atom. The inscribed Fermi sphere accommodates 4.19 electrons per atom and does not touch the zone boundaries. The larger volume of the -phase second zone in comparison with the -phase zone implies that the -structure can accommodate more electrons and thus the solubility of some transition elements is greater in the - phase than in the -phase. 1.5.4 Representative examples of Ti–X and Zr–X phase diagrams In this section representative examples of a few types of Ti–X and Zr–X binary equilibrium phase diagrams will be introduced: Ti–Zr, Ti–Mo, Ti–V, Ti–Cr, Ti–Al, Zr–Nb, Zr–Fe, Zr–Sn, Zr–Al, Ti–N,Zr–H and Zr–O. The phase diagrams presented here are based on those appearing in Massalski et al. (1992). Subsequent updates have been published in respect of some of these binary systems. These updates have been referred to at appropriate places.

30


The Ti–Zr system is an example of an – isomorphous system, while Ti–Mo and Ti–V constitute important examples of so called -isomorphous systems and form the basis of several commercial and + alloys. The Ti–Cr system is a typical -eutectoid system, while the -stabilizer related Ti–Al system is pertinent to several technical and + alloys. The Zr–Nb system, which relates to the important family of commercial Zr–Nb alloys, is a -monotectoid system. The Zr–Fe phase diagram exemplifies a -eutectoid system. The Zr–Sn and Zr–Al systems exhibit -phase stabilization. The former is very relevant with regard to important technical Zr alloys such as zircaloys, while the latter is germane to the Zr 3 Al intermetallic phase which has been considered as a potential nuclear reactor structural material. In all these cases, X is a substitutional solute. In the Ti–N, Zr–H and Zr–O systems, all of which are of technological importance, X is an interstitial solute. The Ti–Zr system (Figure 1.6) appears to be a truly isomorphous system, though perhaps not as close to an “ideal solution” situation as the Zr–Hf system. The equilibrium phases occurring in the Ti–Zr system are the liquid (L), Ti Zr, Ti Zr, -Ti, -Ti, -Zr and -Zr. Apart from these, the metastable (martensite) and -phases are also encountered. The special points of the Ti–Zr system are listed in Table 1.7 (Murray 1987, Massalski et al. 1992.)

Weight per cent Zr 0

10 20

30

40

50

60

70

80

90

100

2270

2128 K 2070

1943 K

L

1870

Temperature (K)

1813 K 1670 1470

(β-Ti, β-Zr)

1270

1155 K

1136 K

1070 870

~878 K (α-Ti, α-Zr)

670 0

Ti

10

20

30

40

50

60

Atom per cent Zr

Figure 1.6. Equilibrium phase diagram for the Ti–Zr system.

70

80

90

100

Zr


31

Table 1.7. Special points of the Ti–Zr system. Phase reaction

Type of reaction

L Ti Zr L Ti L Zr Ti Zr Ti Zr Ti Ti Zr Zr

Congruent Melting Melting Congruent Allotropic Allotropic

Temperature (K)

Composition (at.% Zr)

1813 ± 15 1943 2128 878 ± 10 1155 1136

38 ± 2 0 100 52 ± 2 0 100

Weight per cent Mo 0 10 20

30

40

50

60

70

80

90

100 2896 K

2870 2670

L

2470

Temperature (K)

2270 2070 1943 K

1870 1670

(β-Ti, Mo)

1470 1270

1155 K

~1123 K

1070

~968 K ~12 (α-Ti)

870 670 0

Ti

10

20

30

40

50

60

Atom per cent Mo

70

80

90

100

Mo

Figure 1.7. Equilibrium phase diagram for the Ti–Mo system.

In the Ti–Mo system (Figure 1.7), the equilibrium solid phases that are encountered are: the bcc (-Ti, Mo) solid solution, in which Ti and Mo are completely miscible above the allotropic transformation temperature of Ti (1155 K), the hcp -Ti (Mo) solid solution in which the solubility of Mo is restricted (maximum of about 0.4 at.%), -Ti, -Ti and Mo. This system exhibits a miscibility gap in (-Ti, Mo) and a monotectoid reaction: -Ti) (-Ti) + (Mo) (Terauchi et al. 1978), the monotectoid temperature being about 968 K. The metastable martensite (hcp and orthorhombic ) and -phases also occur in

32


Table 1.8. Special points of the Ti–Mo system. Phase reaction

Type of reaction

L Ti L Mo Ti Mo Ti + Mo Ti Ti + Mo Ti Ti

Melting Melting Critical Monotectoid Allotropic

Temperature (K)

Composition (at.% Mo)

1943 2896 ∼ 1123 ∼ 968 1155

0 100 ∼ 33 (12) (0.4) ∼ 60 0

the Ti–Mo system. The special points of the Ti–Mo system are shown in Table 1.8 (Murray 1987, Massalski et al. 1992). The equilibrium phase diagram of the Ti–V system (Figure 1.8) also shows a miscibility gap in the bcc (-Ti, V) phase and a monotectoid reaction occurring at 948 K: (-Ti) (-Ti) + (V) (Nakano et al. 1980). Above 1155 K, Ti and V are completely miscible in the (-Ti, V) solid solution. The solubility of V in the hcp (-Ti) phase is restricted, with a maximum of 2.7 at.% V. The metastable phases, martensite ( or , depending on the V content) and , are

Weight per cent V 0

10

20

30

40

50

60

70

80

90

100

2183 K

2170 L

Temperature (K )

1970

1943 K 1878 K

1770

1570

(β-Ti, V) 1370

1170

1155 K

1123 K 948 K

970

(α-Ti) 770 0

Ti

10

20

30

40

50

60

Atom per cent V

Figure 1.8. Equilibrium phase diagram for the Ti–V system.

70

80

90

100

V


33

Table 1.9. Special points of the Ti–V system. Phase reaction

Type of reaction

L Ti V L Ti LV Ti V Ti + V Ti Ti + V Ti Ti

Congruent Melting Melting Critical Monotectoid Allotropic

Temperature (K)

Composition (at.% V)

1878 1943 2183 1123 948 1155

32 0 100 ∼ 50 (18) (2.7) (∼ 80) 0

also encountered in this system. The special points pertinent to the Ti–V system are listed in Table 1.9 (Murray 1987, Massalski et al. 1992). Subsequently, an update has been published by Okamoto (1993a) with regard to the Ti–V phase diagram. Figure 1.9 shows the Ti–Cr equilibrium phase diagram. The equilibrium condensed phases encountered are the liquid (L), the bcc (-Ti, Cr) solid solution, the hcp (-Ti) solid solution, the topologically close packed intermetallic phases -TiCr2 , -TiCr2 and -TiCr2 , and, of course, -Ti, -Ti and Cr. In a narrow temperature range below the congruent melting temperature, Ti and Cr are completely Weight per cent Cr 0

10

20

30

40

50

60

70

80

90

100

2270 2136 K 2070 L

1943 K

Temperature (K)

1870 1683 K

1643 K

1670 ~1543 K (β -Ti, Cr)

1470

γ-TiCr2

~1493 K

β -TiCr2 1270 1155 K (α -Ti)

1070

~1073 K 940 K

α -TiCr2

870 0

Ti

10

20

30

40

50

60

Atom per cent Cr

Figure 1.9. Equilibrium phase diagram for the Ti–Cr system.

70

80

90

100

Cr

34


Table 1.10. Special points of the Ti–Cr system. Phase reaction

Type of reaction

L Ti Cr L Ti L Cr Ti TiCr 2 Ti Ti + TiCr 2 Ti + TiCr 2 TiCr 2 TiCr 2 TiCr 2 TiCr 2 TiCr 2 + Cr Ti Ti

Congruent Melting Melting Congruent Eutectoid Peritectoid Unknown Eutectoid Allotropic

Temperature (K)

Composition (at.% Cr)

1683 ± 5 1943 2136 ± 20 1643 ± 10 940 ± 10 ∼ 1493 ∼ 1543 ∼ 1073 1155

44 0 100 ∼ 66 (12 5 ± 0 5) (0.6) (∼ 63) (39) (∼ 63)(∼ 65) ∼ 65 to 66 (∼ 65) (∼ 66) (96) 0

miscible in the (-Ti, Cr) phase. The maximum solubility of Cr in the (-Ti) phase is 0.6 at.%. The martensitic and the -phase also form in this system. The special points germane to the Ti-Cr system are presented in Table 1.10 (Murray 1987, Massalski et al. 1992). In the Ti–Al equilibrium phase diagram, (Figure 1.10), the solid phases that appear are: the bcc (-Ti) and the hcp (-Ti) solid solutions, the ordered intermetallic phases, Ti3 Al (also referred to as 2 ), TiAl (also referred to as ), TiAl, Weight per cent Al 0

10

20

30

40

50

60

70

80 90 100

1970 1943 K

L

1770 ~1558 K

Temperature (K)

(β-Ti)

1570

δ

~1398 K

TiAl 1370

TiAl3 Ti3Al

1170

TiAl2

1155 K

α-TiAl3

(α-Ti)

970

938 K

933 K

(Al) 770 0

Ti

10

20

30

40

50

60

70

Atom per cent Al

Figure 1.10. Equilibrium phase diagram for the Ti–Al system.

80

90

100

Al


35

Table 1.11. Special points of the Ti–Al system. Phase reaction

Type of reaction

L Ti Al L + Ti TiAl L + TiAl L + TiAl3 L + TiAl3 Al L Ti L Al Ti + TiAl Ti Ti Ti3 Al Ti Ti3 Al + TiAl TiAl + TiAl2 TiAl2 + TiAl3 TiAl3 TiAl3 Ti Ti

Congruent Peritectic Peritectic Peritectic Peritectic Melting Melting Peritectoid Congruent Eutectoid Peritectoid Eutectoid Unknown Allotropic

Temperature (K)

Composition (at.% Al)

∼ 1983 ∼ 1753 ∼ 1653 ∼ 1623 938 1943 933 ∼ 1558 ∼ 1453 ∼ 1398 1513 ∼ 1423 ∼ 873 1155

11 (53) (47.5) (51) (73.5) (69.5) (71.5) (80) (72.5) (75) (99.9) (75) (99.3) 0 100 (43) (49) (45) ∼ 32 (40) (39) (48) (65) (70) (67) (71.5) (68) (75) 75 0

and TiAl3 , and the (Al) solid solution. The addition of Al to Ti stabilizes the (-Ti) phase relative to the (-Ti) phase. The maximum solubilities of Al in (-Ti) and (-Ti) are about 48 and 45 at.%, respectively while that of Ti in Al is around 0.7 at.%. The phase boundaries for the TiAl2 and phases are yet to be ascertained. The metastable martensitic phase also forms in the Ti–Al system. The special points of this system are indicated in Table 1.11 (Murray 1987, Massalski et al. 1992). Two updates (Okamoto 1993b, 1994) pertaining to the Ti–Al phase diagram have appeared later. The equilibrium phases encountered in the Zr–Nb system are: the liquid (L), bcc (-Zr, Nb), (-Zr) and (Nb) solid solutions and the hcp (-Ti) solid solution. The bcc (-Zr, Nb) solid solution exhibits a miscibility gap and a monotectoid reaction: (-Zr) ←−−→ (-Zr) + (Nb) occurs. The phase diagram (Abriata and Bolcich 1982, Massalski et al. 1992) is shown in Figure 1.11 and the special points pertinent to the system are listed in Table 1.12. The metastable martensite ( ) and -phases form in this system. The Zr–Nb phase diagram has subsequently been updated (Okamoto 1992). The equilibrium Zr–Fe phase diagram (Arias and Abriata 1988, Massalski et al. 1992) is shown in Figure 1.12. The equilibrium phases are: the liquid (L); the bcc terminal solid solution, (-Zr), in which the maximum solubility of Fe is about 6.5 at.%; the hcp terminal solid solution, (-Zr), in which Fe has a maximum solubility of 0.03 at.%; the four intermetallic phases, Zr3 Fe, Zr 2 Fe, ZrFe2 and ZrFe3 ; the high temperature bcc terminal solid solution, (-Fe), in which Zr has a maximum solubility of about 4.5 at.%; the fcc terminal solid solution, (-Fe)

36

Phase Transformations: Titanium and Zirconium Alloys Weight per cent Nb 0

10

20

30

40

50

60

70

80

90

100

2742 K

2770 L

2570

Temperature (K)

2370 2128 K

2170 1970

21.7

2013 K

1770

(β-Zr, β-Nb)

1570 1370

1261 K 60.6

1136 K

1170 970

0.6

893 ± 10 K

(α-Zr)

770 0

10

91.0

18.5

20

30

Zr

40

50

60

70

Atom per cent Nb

80

90

100

Nb

Figure 1.11. Equilibrium phase diagram for the Zr–Nb system.

Table 1.12. Special points of the Zr–Nb system. Phase reaction

Type of reaction

L Zr Nb L Zr L Nb Zr Nb Zr + Nb Zr Zr + Nb) Zr Zr

Congruent Melting Melting Critical Monotectoid Allotropic

Temperature (K)

Composition (at.% Nb)

2013 2128 2742 1261 893 ± 10 1136

21.7 0 100 60.6 (18.8) (0.6) (91.1) 0

which shows a maximum solubility of around 0.7 at.% Zr; and the low temperature bcc terminal solid solution, (-Fe), in which the maximum solubility of Zr is only about 0.05 at.%. Table 1.13 shows the special points relevant to the Zr–Fe system. Amorphous Zr–Fe alloys have been produced over a wide range of compositions by rapid solidification processing. The metastable -phase also forms in this system. An update of the Zr–Fe equilibrium diagram has appeared later (Okamoto 1993c). The assessed Zr–Sn phase diagram (Abriata et al. 1982, Massalski et al. 1992) is shown in Figure 1.13. In this diagram, there appears to be uncertainty regarding


37

Weight per cent Fe 0

10

20

30

50

40

60

70

80

90

100

2270

2128 K

2070

L

1946 K

1870

δ Fe

66.7

1755 K

Temperature (K)

1670 1470

1667 K

90.2 ~99.3 1630 K (γ-Fe)

(β-Zr)

1270 1070

1811 K

~95.5

1610 K

1247 K

1201 K

~6.5 1158 K 1003 K ~24.0

Zr2Fe

4.0 0.03

870

ZrFe2

670

n ag M ns a tr

~573 K

(α-Zr) 470

1048 K

Zr3Fe

1198 K ~99.9 1185 K 1043 K Magnetic trans (α-Fe) ZrFe3 ~548 K Magnetic trans

270 0

10

20

30

Zr

40

50

60

70

80

Atom per cent Fe

90

100

Fe

Figure 1.12. Equilibrium phase diagram for the Zr–Fe system.

Table 1.13. Special points of the Zr–Fe system. Phase reaction

Type of reaction

L Zr L Zr + Zr 2 Fe L ZrFe2 L ZrFe3 + Fe L Fe L + ZrFe2 Zr 2 Fe L + ZrFe2 ZrFe3 Fe L + Fe Zr Zr Zr Zr + Zr 3 Fe Zr + Zr 2 Fe Zr 3 Fe Zr 2 Fe Zr 3 Fe + ZrFe2 ZrFe3 + Fe Fe Fe Fe Fe Fe

Melting Eutectic Congruent Eutectic Melting Peritectic Peritectic Catatectic Allotropic Eutectoid Peritectoid Eutectoid Peritectoid Allotropic Allotropic

Temperature (K) 2128 1201 1946 1610 1811 1247 1755 1630 1136 1003 1158 1048 1198 1667 1185

Composition (at.% Fe) 0 ∼ 24 ∼ 6 5 31 66.7 (90.2) (75) (∼ 99 3) 100 (∼ 25) (66) (33.3) (86.7) (∼ 72 5) (75) (∼ 95 5) (90.8) (∼ 99 3) 0 (4) (0.03) (24) (∼ 6) (31) (∼ 25) (33.3) (26.8) (66) (75) (?) (∼ 99 95) 100 100

38

Phase Transformations: Titanium and Zirconium Alloys Weight per cent Sn 0

10

20

30

40

50

60

70

80

90

100

2470

2261 K

2270

2128 K 2070

L 1865 K

1670

17.0

19.1

(β-Zr)

1600 K

11.8

1415 K

1470

1255 K

(α-Zr)

Zr4Sn

870

ZrSn

1270 1136 1070

Zr5Sn3

Temperature (K)

1870

(β-Sn)

670

~505 K

505 K

470 0

10

Zr

20

30

40

50

60

70

80

Atom per cent Sn

90

100

Sn

Figure 1.13. Equilibrium phase diagram for the Zr–Sn system.

Table 1.14. Special points of the Zr–Sn system. Phase reaction

Type of reaction

L Zr L Zr + Zr 5 Sn3 L Zr 5 Sn3 L Sn L + Zr 5 Sn3 ZrSn2 Zr + Zr 5 Sn3 Zr 4 Sn Zr + Zr 4 Sn Zr Zr Zr Sn Sn

Melting Eutectic Congruent Melting Peritectic Peritectoid Peritectoid Allotropic Allotropic

Temperature (K)

Composition (at.% Sn)

2128 1865 2261 505 1415 1600 1255 1136 286

0 (19.1) (17) (40) 40 100 (79) (40) (66.6) (11.8) (40) (20) (4.9) (20) (7.3) 0 100

most of the liquidus and the entire region between about 30 and 50 at.% Sn. The special points pertaining to the Zr–Sn system are listed in Table 1.14. The metastable martensitic phase forms in this system. The equilibrium phases encountered in the phase diagram of the Zr–Al system (Massalski et al. 1992) shown in Figure 1.14 are: the liquid (L); the bcc (-Zr) and the hcp (-Zr) solid solutions, the ten intermetallic phases, Zr 3 Al, Zr 2 Al, Zr 5 Al3 ,


39

Weight per cent Al 0

20

10

30

40

50

60 70 80 100

2270 2128 2070

L 1753 K

1463 K ZrAl3

ZrAl

Zr4Al3

(α-Zr)

Zr3Al2

1213 K

Zr2Al

11.5

1136 1070

73.5

1261 K

ZrAl2

12.5

Zr2Al3

1270

Zr5Al4

1470

1548 K

1623 K

1523 K

Zr5Al3

26

1853 K

1863 K

49

37

Zr3Al

Temperature (K)

1803 K

1670 22.5

59

39

29.5

(β-Zr)

1918 K

1758 K

1668 K

1870

(Al) 934 K

870 0

Zr

10

20

30

40

50

60

Atom per cent Al

70

80

90

100

Al

Figure 1.14. Equilibrium phase diagram for the Zr–Al system.

Zr 3 Al2 , Zr 4 Al3 , Zr 5 Al4 , ZrAl, Zr 2 Al3 , ZrAl2 and ZrAl3 , and the fcc (Al) solid solution in which the maximum solubility of Zr is about 0.07 at.%. The addition of Al stabilizes (-Zr) relative to (-Zr) and the maximum solubilities of Al in these two phases are about 11.5 and 26 at.%, respectively. The special points of the Zr– Al system are shown in Table 1.15. Subsequently, three updates in respect of the Zr–Al phase diagram have appeared (Murray et al. 1992, Okamoto 1993d, 2002). The equilibrium condensed phases that occur in the binary Ti–N system are: the liquid (L), the terminal bcc solid solution (-Ti), the terminal hcp solid solution (-Ti), and the three stable nitride phases, Ti2 N, TiN and . Both the terminal solid solutions have wide ranges of composition. The dissolved N (-stabilizer) extends the stability regime of the -Ti phase to a temperature (2623 K) much above the melting point of elemental -Ti. Two of the nitride phases, Ti2 N and , are stable over narrow composition ranges while the third, TiN, exhibits stability over an extensive composition range. Figure 1.15 shows the Ti–N equilibrium phase diagram; the special points of this system are listed in Table 1.16 (Massalski et al. 1992). An update pertaining to the Ti–N phase diagram has appeared subsequently (Okamoto 1993e). Figure 1.16 (Zuzek et al. 1990, Massalski et al. 1992) shows the solid phases encountered in the Zr–H phase diagram. These are the bcc terminal solid solution

40


Table 1.15. Special points of the Zr–Al system. Phase reaction

Type of reaction

L Zr L Zr + Zr 5 Al3 L + Zr 3 Al2 Zr 5 Al3 L + Zr 5 Al4 Zr 3 Al2 L Zr 5 Al4 L Zr 5 Al4 + Zr 2 Al3 L + ZrAl2 Zr 2 Al3 L ZrAl2 L ZrAl2 + ZrAl3 L ZrAl3 L + ZrAl3 Al L Al Zr Zr Zr + Zr 5 Al3 Zr 2 Al Zr + Zr 2 Al Zr 3 Al Zr + Zr 3 Al Zr Zr 5 Al3 Zr 2 Al + Zr 3 Al2 Zr 3 Al2 + Zr 5 Al4 Zr 4 Al3 Zr 5 Al4 Zr 4 Al3 + ZrAl Zr 5 Al4 + Zr 2 Al3 ZrAl

Melting Eutectic Peritectic Peritectic Congruent Eutectic Peritectic Congruent Eutectic Congruent Peritectic Melting Allotropic Peritectoid Peritectoid Peritectoid Eutectoid Peritectoid Eutectoid Peritectoid

Temperature (K)

Composition (at.% Al)

2128 1623 1668 1753 1803 1758 1868 1918 1763 1853 934 933 1136 1523 1261 1213 ∼ 1273 ∼ 1303 ∼ 1273 1548

0 (29.5) (26) (37.5) (∼ 37) (40) (37.5) (∼ 39) (44.4) (40) 44.4 (49) (44.4) (60) (∼ 59) (66.7) (60) 66.7 (73.5) (66.7) (75) 75 (99.97) (75) (99.93) 100 0 (22.5) (37.5) (33.3) (12.5) (33.3) (25) (9.2) (25) (11.5) (37.5) (33.3) (40) (40) (44.4) (42.9) (44.4) (42.9) (50) (44.4) (60) (50)

Weight per cent N 0

2

4

6

8

10

15

20

25

3770 3563 K 47.4

Temperature (K)

3270

L

2770

2623 K 20.5

15.2

28

2293 K 12.5 6.2

2270

4.0

1770

1943 K

TiN

(β-Ti) (α-Ti )

1270

23 1155 K

1323 K 33.3 1373 K 30 33 1073 K 39 34 Ti2N 37.5

δ′

770 0

Ti

5

10

15

20

25

30

35

Atom per cent N

Figure 1.15. Equilibrium phase diagram for the Ti–N system.

40

45

50

55


41

Table 1.16. Special points of the Ti–N system. Phase reaction

Type of reaction

Temperature (K)

Composition (at.% N)

L ←→ Ti L ←→ TiNa L + Ti ←→ Ti L + TiN ←→ Ti Ti + TiN + Ti2 N

Melting Congruent Peritectic Peritectic Eutectoid or Peritectoid Congruent Peritectoid (Probably) Allotropic

1943 ∼ 3563 2293 ± 25 2623 ± 25 1323 ± 60

0 47.4 (4.0) (12.5) (6.2) (15.2) (2.8) (20.5) (23) (30) (33)

∼ 1373 1073 ± 100

33.3 (34) (37.5) (39)

1155

0

TiN ←→ Ti2 Nb Ti2 N + + TiN Ti ←→ Ti a b

Observed under pressure >∼1 MPa. Occurrence if Ti + TiN + Ti2 N equilibrium is eutectoid.

Weight per cent H 1270

1136 K (β-Zr)

Temperature (K)

1070

δ

(α-Zr) 823 K

870

ε

5.93

~37.5

56.7

670

470

270 0

Zr

10

20

30

40

50

60

70

80

Atom per cent H

Figure 1.16. Equilibrium phase diagram for the Zr–H system.

(-Zr), which decomposes eutectoidally at 823 K at a H concentration of 37.5 at.%, the hcp terminal solid solution (-Zr) which exhibits a maximum H solubility of 5.9 at.% at 823 K and the hydride phases (fcc) and (fct). The Zr–O phase diagram (Abriata et al. 1986, Massalski et al. 1992) is shown in Figure 1.17. The equilibrium condensed phases are the liquid (L), the bcc terminal solid solution (-Zr), the hcp terminal solid solution (-Zr) and the oxide phases, -ZrO2−x (cubic, cF12), -ZrO2−x (tetragonal, tP6) and -ZrO2−x (monoclinic, mP12). The special points of the Zr–O system are shown in Table 1.17.

42

Phase Transformations: Titanium and Zirconium Alloys Weight per cent O 0

10

20

30

3270

2983 K L + G

P = 1 atm

2403 K

2070

2338 K

25

10 10.5

35 40

19.5

(α-Zr)

(β-Zr)

62

~1798 K

63.6 66.5

31.2

1670

~1478 K 29.8 1270

1136 K

(α′-Zr)

(α3″-Zr) 870 (α2″-Zr) (α1″-Zr)

~1243 K

29.1

66.7

~773 K 28.6 (α ″-Zr) 4

66.7

α-ZrO2-x

Temperature (K)

2470

~2650 K β-ZrO2-x

2243 K 2128 K

γ -ZrO2-x

L

2870

470 0

10

20

Zr

30

40

50

60

70

Atom per cent O

Figure 1.17. Equilibrium phase diagram for the Zr–O system. Table 1.17. Special points of the Zr–O system. Phase reaction

Type of reaction

L ←→ Zr

Melting

2128

0

L ←→ Zr

Congruent

2403 ± 10

25 ± 1

L ←→ ZrO2−x

Congruent

2983 ± 15

66.6

L ←→ Zr + ZrO2−x L + Zr ←→ Zr

Eutectic

2338 ± 5

40 ± 235 ± 162 ± 1

Peritectic

2243 ± 10

10 ± 0 519 5 ± 2 10 5 ± 0 5 ∼ 66 6 ∼ 66 6 ∼ 100

L + ZrO2−x + G ZrO2−x ←→ Zr + ZrO2−x ZrO2−x ←→ Zr + ZrO2−x ZrO2−x + ZrO2−x + G

Temperature (K)

2983 Eutectoid

ZrO2−x +ZrO2−x +G

∼ 1798

Composition (at.% O)

∼ 1478

63 6 ± 0 4 31 2 ± 0 5 66 5 ± 0 1 ∼ 66 5 29 8 ± 0 5 ∼ 66 5

∼ 2650

∼ 66 6 ∼ 66 6 ∼ 100

∼ 1478

∼ 66 6 ∼ 66 6 ∼ 100

ZrO2−x ←→ ZrO2−x

Congruent

∼ 2650

66.6

ZrO2−x ←→ ZrO2−x

Congruent

∼ 1478

66.6

Zr ←→ Zr

Allotropic

1136

0


1.6

43

NON-EQUILIBRIUM PHASES

1.6.1 Introductory remarks Phases such as the -, - and intermetallic phases mentioned earlier are equilibrium phases and the corresponding phase fields are delineated in equilibrium phase diagrams of the type described in the previous section. However, non-equilibrium or metastable phases, as distinct from equilibrium phases, are quite important in respect of many alloy systems, including those based on Ti and Zr. Equilibrium phase diagrams are usually developed by deducing the initial states of alloys which have been quenched from different temperatures to room temperature. But the quenching process may lead to the formation of non-equilibrium phases. Two important examples of such non-equilibrium phases in Ti–X and Zr–X systems are the martensite and the athermal -phases. Both these phases are formed through athermal displacive transformations. It will be seen in a later chapter that one way of classifying phase changes is to divide them into two broad classes: reconstructive and displacive (Roy 1973, Christian 1979, Banerjee 1994). Transformations of the former kind involve breaking of the bonds of atoms with their neighbours and re-establishment of bonds to form a new configuration in place of the pre-existing one. Such a process requires atomic diffusion comprising random atomic jumps and disturbs atomic coordination. Atomic movements in displacive transformations, on the other hand, can be brought about by a homogeneous distortion, by shuffling of lattice planes, by static displacement waves or by a combination of these. Cooperative movements of a large number of atoms in a diffusionless process accomplish the structural change in displacive transformations. Unlike the diffusional atomic jumps which are thermally activated, the displacive movements do not require thermal activation and cannot, therefore, be suppressed by quenching. A structural transition involving periodic displacements of atoms from their original positions can be described in terms of a displacement wave and the introduction of a displacement wave in the parent lattice requires coordinated atom movements in an athermal process; the athermal martensitic and -transformations can, respectively, be described in terms of long wavelength and short wavelength displacement waves (Banerjee et al. 1997). In the present chapter brief accounts of the martensite and the -phases and of phase separation in the -phase will be provided with reference to Ti–X and Zr–X alloys. A detailed coverage in respect of the same will be found in three of the subsequent chapters. The martensitic transformation, which is diffusionless and involves cooperative atom movements, proceeds by the propagation of a shear front at a speed that approaches the speed of sound in the material, leading to the formation of the metastable martensite phase. This transformation occurs in many alloy

44


systems, including Ti–X and Zr–X systems, in which the major component exhibits allotropy. The -phase, which is an equilibrium phase in Group 4 metals (Ti, Zr, Hf) at high pressures, forms in several alloys based on these metals and also in many other bcc alloys at ambient pressure as a metastable phase. On rapidly quenching Ti–X and Zr–X alloys, X being an -stabilizing element, from the -phase field, the martensite phase, m , which has the hcp structure, is obtained. The situation is somewhat different when X is a -stabilizing element, such as a transition metal. During the process of rapid cooling from the -phase field, when a composition-dependent temperature (known as the martensitie start or Ms temperature) is crossed, the bcc -phase commences to transform spontaneously by the martensitic mode to the martensite phase m whose structure may be hcp ( ) or orthorhombic ( ), depending on the alloy composition. However, in the case of these alloys, another athermal process, namely, that associated with the formation of the athermal -phase, competes with the martensitic process. At any temperature compatible with the formation of both m and -phases, there is a narrow range of composition (or electron to atom ratio), just beyond the martensite formation regime, over which the athermal -phase forms from the parent -phase. If a s temperature, akin to the Ms temperature, is conceived as being associated with the start of athermal -phase formation, then one may visualize that the s locus lies above the Ms locus in the narrow composition range referred to above, if temperature is plotted against composition. In the composition regime of martensite formation, which lies to the left of this narrow range, the Ms locus lies above the s locus. Even though the -phase appears athermally on rapid quenching from the -phase field only over a narrow range of electron to atom ratio, this phase occurs over a broader composition range as a precipitation product of -phase decomposition. The typical structures exhibited by rapidly -quenched binary Ti–X or Zr–X alloys, X being a -stabilizing element, are indicated in the schematic shown in Figure 1.18. Beyond the + region (where these two phases coexist), the -phase is retained in a metastable (susceptible to decomposition on ageing) or stable manner on quenching. It may be noted that similar values of the electron to atom ratio (∼ 4 15) characterize the limit of the stability of the bcc -phase with respect to either of the two athermal transformations (Collings 1984). 1.6.2 Martensite phase 1.6.2.1 Crystallography The phenomenological crystallographic theories of the martensitic transformation are based on the concept that the interface between the martensite and the parent phases is macroscopically invariant. The central theme of these theories is that the total macroscopic shear consists of three components: (a) the lattice shear or the

Temperature


45

Ms

ωs

Concentration of X

I

II

III

IV

Figure 1.18. Schematic showing the Ms and s loci for a binary Ti–X or Zr–X system, X being a -stabilizing element, on rapid cooling from the -phase field. Region I corresponds to martensite (m ) formation; in regions II and III, the -phase co-exists with the athermal - and the aged -phases, respectively; in region IV only the -phase occurs in a metastable or stable state.

Bain strain which brings about the necessary change in the lattice (e.g. bcc to hcp); (b) a lattice invariant inhomogeneous shear which provides an undistorted plane; and (c) a rigid body rotation to ensure that the undistorted habit plane is unrotated as well. The inhomogeneous shear accompanying the martensitic transformation is instrumental in generating the martensite substructure which, in most cases, is too fine to be resolved under the light microscope. Transmission electron microscopy (TEM) techniques have been extensively used for resolving this substructure and for obtaining information regarding the orientation relationship, the habit plane and the nature of the inhomogeneous strain for individual martensite crystals. A unique feature of the → martensitic transformation in Ti and Zr is that the necessary lattice strains approximately satisfy the invariant plane strain condition. Because of this, the magnitude of the lattice invariant shear is comparatively small and it is relatively simple to characterize the substructure of the martensite in Ti–X and Zr–X alloys. There are a number of choices for relating the lattices of the parent () and product ( ) phases. The correct choice of lattice correspondence is generally made by selecting the one which involves the minimum distortion and rotation of the lattice vectors. In the case of the transformation in Zr, it has been suggested (Burgers 1934) that the 011 plane forms the basal plane 0001 , while the ¯ and 111 ¯ directions lying on that plane correspond to the close packed 111 ¯ ¯ close packed 1120 directions. This accounts for four of the six 1120 ¯ directions. directions; the remaining two are derived from the 100 and 100

46


Having chosen this lattice correspondence, the next step would be to determine the magnitudes of the strains which would deform the distorted hexagonal structure into a regular one, having lattice parameters consistent with those of -Zr. If ao a and c refer to the lattice parameters of the - and -phases, respectively, the magnitudes of two of the principal lattice distortions, 1 (along 100 ) and 2 ¯ ) are given by 1 = a/ao and 2 = 3/2 21 a/ao . The distortion 3 along (along 011

011 is 1/21/2 a/ao where = c/a. On substituting the values for the lattice parameters at the transformation temperature, the magnitudes of the principal strains for pure Zr are seen to be as follows: 2% expansion along 011 , 10% expansion ¯ and 10% contraction along 100 . The situation is analogous in the case along 011 of Ti and the corresponding principal strains are 1% expansion, 11% expansion and 11% contraction, respectively, along the aforementioned directions. A pair of planes remains undistorted under the action of a homogeneous lattice strain if and only if one of the principal strains is zero and the other two are of opposite signs (Wayman 1964). A special feature of the martensitic bcc to hcp transformation in Ti and Zr is that the principal strain along the 011 direction is very small and the other two principal strains are of opposite signs. If the principal strain along the 011 direction, 3 , were zero, the lattice shear would have left a plane undistorted. Since 3 is very small in the case of Ti and Zr, it is not unreasonable to treat the transformation with the approximation that 3 is zero (Kelly and Groves 1970). It has been reported (Bagaryatskii et al. 1959, Flower et al. 1982) that the normally observed hcp ( ) structure of the martensite is distorted to an orthorhombic structure ( ) in many Ti–X systems, X being a transition metal, when the martensite is supersaturated beyond a certain limit. The orthorhombic distortion increases with increasing solute content. It has been noticed that the deformation induced martensite, mentioned later in this section, almost invariably has an orthorhombic structure (Williams 1973). This is not surprising when one considers the fact that this type of martensite can occur only in alloys which are so enriched in -stabilizing solutes that they are not transformed on -quenching. It has been demonstrated (Otte 1970) that the → transformation ¯ ¯¯ ¯ 111 ≡ 2112 involves the activation of the shear systems 112 2113 and ¯ 111 ≡ 1011 ¯ ¯¯ 101 [2113 . The habit plane associated with this transformation has been found to be very close to 334 (Williams 1973, Shibata and Ono 1977), although in some -stabilizing solute enriched Ti–X alloys 344 habit has also been reported (Liu 1956, Gaunt and Christian 1959, Hammond and Kelly 1970). The orientation relationship between the - and -phases has been ¯ > (approximately), which observed to be: 011 0002 ; < 111¯ > < 1120 is consistent with the approximate orientation relation deduced by Burgers with regard to the bcc → hcp transformation in elemental Zr (Burgers 1934).


47

In the case of the orthorhombic martensite, the orientation relationship has been reported to be: 100 and 010 inclined by about 2o from 001 and ¯ > (Hatt and Rivlin 1968). < 110 > , respectively; 001 < 110 1.6.2.2 Transformation temperatures The martensitic transformation is characterized phenomenologically by the assignment of several temperatures. The most common among these are: Ms , the temperature at which martensite starts forming during quenching; Mf , the temperature at which the transformation is completed; s , the temperature at which the m → reverse transformation starts during up-quenching (in Ti and Zr alloys); and To , the temperature at which the free energies of the parent and martensite ( and m ) phases are equal. If Ms and s are very close to each other, it is indicated that the driving force for the transformation is small and also that To can be taken to be the mean of these two temperatures (Collings 1984). A thermodynamic analysis (Kaufman 1959) of the → transformation in several Ti and Zr base alloys has shown that the To temperatures are about 50 K higher than the experimentally observed Ms temperatures. This implies that the supercooling (To –Ms ), necessary to initiate the martensitic transformation in these systems is relatively low. The change in free energy accompanying the transformation at the Ms temperature is significantly lower as compared to that in ferrous systems. At the Ms temperature, the chemical driving force necessary to start a martensitic reaction depends on the shear modulus of the alloy at the transformation temperature, the magnitude of the homogeneous shear associated with the transformation and the magnitude of the inhomogeneous shear. The strain energy associated with martensite formation is determined by the homogeneous lattice strains and the shear modulus while the surface energy corresponds to the energy of the interface between the parent and the product lattices. If the austenite–martensite reaction in ferrous systems is compared with the → transformation in Ti and Zr base alloys, it is found that although there is not much difference in the homogeneous strain values in the two cases, the shear modulus of ferrous alloys at the transformation temperature is much higher. Again, the energy associated with the parent–martensite interface can also be expected to be much smaller in the case of Ti- and Zr-based alloys because only a small amount of inhomogeneous shear is necessary to make the total strain an invariant plane strain in the case of the → transformation. These considerations indicate that the “back stress”, which arises from the strain and surface energies opposing martensite formation, is much smaller in Ti- and Zr-based alloys as compared to ferrous alloys. This explains why a small driving force is adequate for initiating martensite formation in the former. The chemical driving force which balances

48


the “back stress” at the Ms temperature can be assisted by external stress, leading to stress-assisted or stress-induced or deformation-induced martensite formation. The Ms temperature is composition dependent. Again, the measured Ms temperature for a given alloy composition may exhibit a dependence on the rate of cooling (Jepson et al. 1970). In -stabilized Ti–X and Zr–X alloys, Ms increases with increasing solute content and may lie a little below the + / transus; in -stabilized alloys, Ms decreases with increasing concentration of X and always lies in the ( + ) field (Collings 1984). In dilute alloys of the latter type, the Ms temperature is relatively high and water quenching may not be sufficiently rapid for completely suppressing thermally activated atom movements, leading to some segregation of the solute atoms prior to the transformation and consequently to the retention of some -phase. If the solute content in the alloy increases, the Ms temperature decreases and the diffusional contribution is inhibited, with the result that a full transformation to the martensite phase comes about. It may be mentioned here that the quench rates necessary to achieve the structural transformation while preserving compositional homogeneity depend strongly on the nature of the alloying element, X, or more specifically, on its diffusion kinetics in the -phase. A quench rate that is adequate when X is an early transition metal such as V, Nb or Mo, may not be so when X is a late transition metal like Fe, Co or Ni. This is so because metals belonging to the latter category diffuse much faster in the -phase; for instance, in the context of diffusion in -Ti at 1273 K, it may be noted that the diffusion coefficients of Co and Mo are in the ratio 200:1 (Collings 1984). As the solute concentration increases further, a stage is reached where the Mf temperature drops below the temperature of the quenching bath; in this situation, the retention of some untransformed -phase again becomes feasible. 1.6.2.3 Morphology and substructure If dilute Ti–X and Zr–X alloys are quenched from the -phase field, maintaining an adequately fast cooling rate, one generally obtains a hcp martensite phase ( ) which is known as lath or packet or massive martensite and consists of relatively large, irregular packets or “colonies” which are populated by nearparallel arrays of much finer platelets or laths. No retention of the -phase occurs in a lath martensite. As the solute content increases, the average packet size and the average lath size decrease. Beyond a certain level of solute concentration, which depends on the nature of the solute, a transition occurs in the martensite morphology, resulting in the formation of plate or acicular martensite. In contrast to the arrangement of near-parallel units in the lath morphology, the martensite units form in various intersecting directions in the plate or acicular structure. Another important difference between the lath and the plate morphologies is that the size distribution in the latter case is much broader than in the former. This is essentially


49

due to the fact that in the plate morphology the martensite units continuously partition the parent -grain and as a result of this the space available for the growth of the plates belonging to the subsequent generations gets more and more limited. The transition from lath to plate morphology is not abrupt and the two may coexist over some range of composition. When the solute concentration is sufficiently high, the martensitic transformation may be incomplete and some -phase, which is usually trapped between the platelets of the acicular martensite, may be retained. Not far removed from the “ plus acicular martensite” quenched structure is the Widmanstatten arrangement consisting of groups of -phase needles lying with their long axes parallel to the 110 planes of the retained -phase. The term “substructure” of a martensite generally refers to the structure within the martensite unit as revealed under the transmission electron microscope (TEM). This substructure arises from (a) the lattice invariant component of the transformation strain which may be slip, twin or a combination of both; and (b) the post-transformation strain resulting from the accommodation effect. There has been considerable interest in characterizing the internal structure of martensite plates for determining the nature of the inhomogeneous shear participating in the transformation process, as envisaged in the phenomenological theory of the martensitic transformation. For this it is necessary to be able to identify and separate the inhomogeneities introduced by matrix constraints from those produced by the lattice invariant component of the transformation strain. Such a separation is not straightforward. In a twinned martensite plate, a set of transformation twins is expected to appear periodically at almost equal intervals within the plate; the ratio of the thicknesses of the twinned and the matrix portions should be consistent with the value predicted by the theory and the specific variant of the twin plane should be consistent with the observed habit plane. When a set of twins in a martensite plate satisfies all these conditions, the twins are taken to be transformation twins. In a dislocated martensite crystal, it is more difficult to separate the transformation induced dislocations from those introduced by post-transformation stresses. A rule of the thumb appears to be that only those dislocations which are arranged in regular arrays and are observed very frequently may be taken to have been produced by the inhomogeneous shear. Generally, a transition from the dislocated to the twinned substructure is found to occur with increasing concentration of alloying elements. 1.6.3 Omega phase 1.6.3.1 Athermal and isothermal It has been mentioned earlier that under ambient pressure, the -phase can occur in a metastable manner in alloys in which the -phase is stabilized with respect to the martensitic → m transformation. The composition range over which this phase may be encountered is a characteristic of the alloy system under consideration. It has also been indicated that this phase can be obtained either by rapidly quenching

50


from the -phase field (athermal ) or as a product of thermally activated -phase decomposition (isothermal or aged ). The athermal → transformation is displacive, diffusionless and of the first-order and the -phase so obtained has a composition very close to that of the -phase. The thermally activated transformation, on the other hand, is accompanied by solute rejection by diffusional processes from the to the -phase and is thus partially replacive in nature. The athermal → transformation cannot be suppressed even by extremely rapid quenching and is completely and continuously reversible with negligible hysteresis. The special characteristics of this transformation also include the appearance of an extensive diffuse intensity distribution in diffraction patterns, with the maximum intensity located close to the positions of ideal -reflections, as a precursor to the transformation event and the stability of the dual phase + structure with extremely fine (∼1–4 nm) particles distributed in the -matrix along 111 directions (Banerjee et al. 1997). The number density of the particles is extremely large and this fact lends support to the contention that the transformation does not involve long range diffusion. The volume fraction of the isothermal -phase forming in the -matrix is a function of the reaction time. This dependence of the volume fraction on time arises essentially due to the diffusion controlled partitioning of the solute between solute lean and solute rich regions. The solute lean regions are eventually transformed to the -phase. The composition of the isothermal -phase corresponds to the maximum solubility of the solute in the -phase. Thus after prolonged ageing at temperatures lower than about 770 K, a metastable + state is attained, characterized at a given temperature by a fixed volume fraction and composition of each of the and terminal points (Hickman 1969). After sufficiently long ageing periods at 720–770 K, -phase precipitation can be expected. An early model of isothermal -phase development visualized an initial structural transformation of the lattice into and (as in the case of the athermal -phase in its pertinent composition regime), followed by an exchange of solute and solvent atoms across the / interface (Courtney and Wulff 1969). It has subsequently been suggested that initially a composition fluctuation occurs and this is followed by a structural → transformation within a solute lean zone, triggered by a longitudinal phonon with a 2/3 111 wave vector; it is the instability of the bcc lattice with respect to this disturbance that is responsible for the athermal transition (de Fontaine et al. 1971). 1.6.3.2 Crystallography The crystal structure of the -phase has been described in an earlier section. The orientation relationship between the and -phases has been determined by a large number of investigators and has been unanimously accepted as: 111 0001 ;


51

¯ 1210 ¯ 110 . It has been found that this orientation relationship is valid both for the athermal -phase and the isothermal -phase (Williams 1978). This relationship implies that there are four possible crystallographic variants of the -structure, depending on which one of the 111 planes is parallel to the (0001) plane. Again, for the same variant of the -structure, there are three 110 directions so that in all 12 variants of the -structure are possible. But since the basal plane of this structure has six-fold symmetry, the three variants for a given 111 plane will appear identical and, therefore, the contribution from only four variants will be seen in selected area diffraction (SAD) patterns. The lattice parameters, a and c of the structure and a of the (bcc) structure, are related as follows: a =

√

√ 2a c = 3/2a

1.6.3.3 Morphology Precipitates of the athermal -phase that evolve during rapid -quenching are very fine (1%), as is generally the case when the solute is a 3d-transition metal like V, Cr, Mn or Fe, the minimization of elastic strains in the cubic matrix dictates a cubic morphology (Hickman 1969, Blackburn 1970). 1.6.3.4 Diffraction effects Pronounced diffuse scattering has been observed in electron, X-ray and neutron diffraction patterns prior to the formation of the -phase in all -forming systems. These diffuse intensity patterns are closely associated with the non-diffuse (sharp) reflections corresponding to the crystalline -phase. In view of this close association, the diffuse intensity distribution has been attributed to non-ideal -structures. It has been mentioned earlier that the ideal hexagonal -structure is obtained when the parameter z has the value zero and the non-ideal trigonal -structure results if 0 < z < 1/6. Selected area electron diffraction patterns obtained from the truly athermal -phase are characterized by sharp spots and straight lines of intensity while broad reflections and either straight or curved diffuse lines of intensity

52


(diffuse streaking) originate from the “diffuse” -phase (Collings 1984). A model proposed by Sass and co-workers (Dawson and Sass 1970, McCabe and Sass 1971, Balcerzak and Sass 1972) envisages an ensemble of -particles, 1–1.5 nm in diameter and 1.5–2.5 nm apart, arranged in rows along 111 directions. According to this model, clusters of such rows contribute to the sharp spots and straight lines of intensity, while the broad reflections and diffuse streaking arise from either individual rows of particles or isolated particles. It has been demonstrated that a transition from diffuse to sharp -reflections occurs in -quenched specimens in response to either decreasing solute content (Sass 1972) or decreasing temperature (de Fontaine et al. 1971). In both cases, curvilinear lines of diffuse intensity become straight and well defined. It has been pointed out (Williams 1973) that since the diffuse streaking tends to coincide with the positions of the -reflections when they are present, compositionwise there is no sharp line of demarcation separating the regions of athermal and diffuse . A soft phonon mechanistic model of the -phase reaction (de Fontaine et al. 1971) has been able to provide a rationalization, in terms of lattice dynamics, for the temperature and composition dependences of the athermal and diffuse -phases. After examining electron diffraction patterns belonging to several zones and considering the symmetry of the reciprocal lattices, de Fontaine et al. (1971) have constructed a three-dimensional model of the diffuse intensity which is distributed on quasi-spherical surfaces centred around the octahedral sites of the reciprocal of the bcc -lattice. These spheres of intensity touch all the 111 faces of the octahedra surrounding them. When this intensity distribution in the reciprocal space is sectioned to reveal the diffuse intensity pattern in a plane corresponding to any zone, the pertinent shifts of the diffuse intensity maxima from the positions of ideal -reflections and asymmetry in intensity distribution are manifested. The lattice dynamical model for phase stability, with special reference to the quenched -phase, has been further developed by Cook (1975). It does appear that the 23 111 soft mode, interacting with a lattice of composition and temperature dependent relative stability, is responsible not only for athermal but also for diffuse , which represents in varying degrees, dynamical fluctuations between the - and -phases (Collings 1984). 1.6.4 Phase separation in -phase Below a transus representing the upper boundary of a region in the equilibrium phase diagram known as a miscibility gap, a previously homogeneous single phase solid solution decomposes into a thermodynamically stable aggregate of two bcc phases, one solute lean and the other solute rich, designated respectively as the 1 - and 2 -phases. Two ideal examples of systems where such a → 1 + 2 decomposition occurs are the Zr–Nb and Zr–Ta systems, both of which exhibit


53

the monotectoid reaction 1 → + 2 . The equilibrium phase diagrams for these systems have many points of similarity with those of Zr–X and Ti–X (X being a transition metal) eutectoid systems; in the case of the latter, Ti is replaced by Zr and the intermetallic phase Tim Xn replaced by 2 . A bimodal free energy (G) versus solute concentration (X) curve is associated with either of the Zr–Nb and Zr–Ta systems, representing the occurrence, in equilibrium, of two solid solutions. This situation is somewhat different from that representing the coexistence of say, the -and -phases in equilibrium. While the latter situation is described in terms of two independent free energy parabolas, equilibrium phase separation, with the absence of structural change, has to be described by a continuous curve with two minima, separated by an intervening maximum, for which the second derivative of free energy with respect to solute concentration is negative. Although above the monotectoid temperature, both 1 and 2 are equilibrium phases, the former ceases to be an equilibrium phase below this temperature. However, the 1 -phase has been found to occur in a metastable manner at temperatures close to but lower than the monotectoid temperature in the Zr–Nb (Banerjee et al. 1976, Menon et al. 1978) as well as the Zr–Ta (Mukhopadhyay et al. 1978, Menon et al. 1979) systems. Phase separation in the -phase has also been reported in some Ti–X systems (X: Cr, V, Mo, Nb). In situations where the temperature (Williams et al. 1971) or the solute concentration (Williams 1973) is too high to be conducive for -phase precipitation, a solute lean bcc phase, designated as separates from the -phase. The → + phase separation reaction can be considered to be a clustering reaction characteristic of alloy systems which exhibit positive heats of mixing (Chandrasekaran et al. 1972) or similar manifestations of a tendency for the alloying constituents to unmix. It is interesting to note that -stabilizing elements such as Al, Sn and O, when added to Ti–V and Ti–Mo alloys in sufficient quantities, appear to increase the stability of the bcc lattice in that -phase formation is suppressed in favour of -phase separation (Williams 1971). Thus these solutes, which are certainly not -stabilizers in the conventional sense, can be regarded as stabilizers of the bcc lattice against the instability. Ageing in the + phase field, which lies just outside the + phase field, would eventually result in -nucleated -phase precipitation.

1.7

INTERMETALLIC PHASES

1.7.1 Introductory remarks A large number of intermetallic phases are encountered in binary Ti–X and Zr–X systems and these exhibit a variety of crystal structures. Many of these structures

54


are derived from three simple crystal structures, namely, face centered cubic (fcc, Al), body centered cubic (bcc, A2) and hexagonal close packed (hcp, A3) structures, which are commonly associated with pure metals and disordered solid solutions. It may be noted that the formation of intermetallic phases does not appear to occur in these binary systems when X is an alkali or alkaline earth metal (with the exception of Be) or a transition metal belonging to Group 3 or Group 4. Likewise, no intermetallic phases form when X is a rare earth (RE) element. Ti–RE and Zr–RE phase diagrams are normally characterized by the absence of intermetallic phases, limited mutual solubilities in the solid state, and quite often, a miscibility gap in the liquid state. In general, in the intermetallic phase Zrm Xn , X is a transition metal from Group 5 (only V) to Group 10 or a simple (non-transition) metal from Group 11 to Group 16. In a similar manner in the intermetallic phase, Tim Xn , X is a transition metal from Group 6 (only Cr) to Group 10 or a non-transition metal from Group 11 to Group 16. In both Tim Xn and Zr m Xn families, X is more often a non-transition metal than a transition metal. However, this does not imply that the incidence of X being a transition metal is infrequent: close to a hundred binary intermetallic phases of this type have been reported. Almost all the important binary intermetallic phases that have been observed in Ti–X and Zr–X alloys, together with hydrides, borides, carbides, nitrides, oxides, phosphides and sulphides have been listed in Tables A1.1 and A1.2, respectively. The composition range, space group, Pearson symbol and strukturbericht designation associated with each of these phases have been incorporated in these tables. The nomenclatures of crystal structures, in terms of their strukturbericht designations and the corresponding Pearson symbols, are listed in Table A1.3 for ready reference. A survey of Table A1.1 and Table A1.2 shows that more than 80% of Tim Xn and Zrm Xn type intermetallic phases are almost equally distributed among three crystal classes: cubic, tetragonal and hexagonal. The orthorhombic system comes next (∼15%) while less than 5% belong to the rhombohedral and monoclinic systems. The unit cells of a majority (∼62%) of these phases are of the primitive type, followed by body centred (∼21%), face centred (∼10%) and base centred (∼7%) cells. The occurrence of body centred unit cells is most common in tetragonal phases, of face centred cells in cubic phases and of base centred cells in orthorhombic phases. The more frequently encountered structures in binary Ti–X intermetallics are the B2 (cP2, CsCl type), C11b (tI16, MoSi2 type), L12 (cP4, AuCu3 type), A15 (cP8, Cr 3 Si type), L1o (tP4, AuCu type) and D019 (hP8, Ni3 Sn type) structures. In the case of Zr–X intermetallics, these are the B2, C11b , C15 (cF24, Cu2 Mg type), D88 (hP16, Mn5 Si3 type), C16 (tI12, Al2 Cu type), C14 (hP12, MgZn2 type) and Bf (oC8, CrB type) structures.


55

Generally speaking, there are many intermetallics that can be put to a variety of uses. For example, there has been a considerable interest in developing strong alloys based on intermetallic phases for structural applications. However, such intermetallics are normally brittle and for this reason, their processing and application are difficult. But it has to be pointed out here that as a class of materials intermetallics, in which atomic bonding is at least partly metallic, tend to be less brittle than ceramics, where atomic bonding is mainly covalent or ionic in nature. Broadly speaking, alloys based on intermetallic phases are hard to deform plastically as compared to pure metals or disordered alloys because of their stronger atomic bonding and the resulting ordered distribution of atoms which gives rise to relatively complex crystal structures. The brittleness of intermetallics generally appears to decrease with increasing crystal symmetry and decreasing unit cell size. In view of this, intermetallics with relatively high crystal symmetry (e.g. cubic, such as B2, D03 , L12 , or nearly cubic, such as L1o , D022 , where a slight tetragonal distortion is present) are thought to have good potential for structural applications (Sauthoff 1996). In the context of Ti–X and Zr–X systems, the intermetallics that have been considered for structural applications include Ti3 AlD019 , TiAlL1o , TiAl3 D022 and Zr 3 AlL12 . Another potential application area pertains to hydrogen storage: certain Ti and Zr bearing Laves phase intermetallics show promise with regard to applications as hydrogen storage materials (Sauthoff 1996). 1.7.2 Intermetallic phase structures: atomic layer stacking The structures of many intermetallic phases can be considered to be formed by the sequential stacking of certain polygonal nets of atoms. These structural characteristics can be readily described by using specific codes and symbols, which can be very useful for a compact presentation and comparison of the structural features of different materials. Various notations have been devised for describing the stacking patterns (Pearson 1972, Ferro and Saccone 1996). Without going into details of these, only the Schlafli notation, P N , will be introduced here. In this notation, P N describes the characteristics of each node in the network in the following manner: the superscript N is the number of P-gon polygons surrounding the node. Thus P = 3 corresponds to a triangle, P = 4 to a rectangle or a square, P = 5 to a pentagon, P = 6 to a hexagon and so on. Some of the very commonly occurring nets are 36 (triangular, T net), 44 (square, S net), 63 (hexagonal, H net) and 3636 (kagome net or K net). These four types of nets are shown schematically in Figure 1.19(a). If a network has nodes which are not equivalent in terms of the polygons surrounding them, the net can be described by listing successively the different corners. For example, if a net is described as 32 434 + 33 42 (2:1), the implication is that in this net there are two types of nodes, 32 434 and 33 42 , and that they occur with a relative frequency of 2:1 (Figure 1.19(b)). A node of the first

56

Phase Transformations: Titanium and Zirconium Alloys T-net

S-net

N-net

K-net

(a)

32434 + 3342(2:1) (b)

Figure 1.19. (a) Schematic representation of a 36 (triangular) T net, a 44 (square) S net, a 63 (hexagonal) H net and a 3636 (kagome) K net of points. (b) The net shown is more complex and contains two types of nodes. This net can be described by the notation 32 434 + 33 42 (2:1). The implication is that these two types of nodes occur with a relative frequency of 2:1. A node of the first type (32 434) is surrounded, in the given order, by two triangles, one square, one triangle and one square, while a node of the second type (33 42 ) is surrounded by three triangles and two squares.

type is surrounded, in the given order, by two triangles, one square, one triangle and one square while a node of the second type is surrounded by three triangles and two squares. A close packed layer of atoms forms a 36 net composed of equilateral triangles. However, not all 36 nets of atoms correspond to close packed layers. To cite an example, the triangles of 36 nets of bcc 110 layers are not equilateral but have angles of 55o , 55o and 70o approximately. In the case of close packing (i.e. a 36 net comprising equilateral triangles), the nodes of one net lie over the centres of


57

the triangles of the nets immediately above and below. Such a situation is not obtained in respect of the stacking of the triangles of the 36 nets of bcc 110 layers (Pearson 1972). The morphologically triangular, 36 (close packed), hexagonal, 63 , and kagome, 3636, nets, together with those made up of squares, are of frequent structural occurrence. A 36 net can be subdivided into a 63 and a larger 36 net (the ratio of number of sites being 2:1) or into a kagome net and a larger 36 net (the ratio of number of sites being 3:1) (Pearson 1972). A primary classification of structures in terms of the stacking of (nearly) planar layers of atoms is quite instructive. For example, numerous structures can be formed by the stacking of T, H or K (triangle– hexagon) layer nets of atoms one over the other sequentially. It is a characteristic of each such layer that it can be positioned about one of three equivalent sites, A, B and C, and this leads to the possibility of varying the stacking sequence and/or the succession of the net types. Moreover, planes of atoms can be constituted of a combination of different layer networks (e.g. hexagonal plus triangular), each of which is occupied by a different chemical species. It is possible to derive a very large number of structure types by permuting the stacking and net sequences. Geometrically close packed (GCP) structures are obtained when the permutation involves only the stacking sequences of the equilateral triangular net. The number of possible structure types may be further increased by chemically ordering the component atoms on the triangular nets. Other structures may be generated by stacking together layer networks of atoms comprising only squares or squares along with triangles, pentagons and/or hexagons. The squares, pentagons or hexagons of one net may or may not be centred by atoms of nets above and below (Pearson 1972). It is interesting to see how several frequently encountered structures in respect of Ti–X and Zr–X intermetallics can be described in terms of the stacking of different types of layer networks of atoms. Before that a brief introduction to topologically close packed (TCP) structures will be provided in view of the fact that quite a few of these intermetallic phases have such structures. Octahedral and tetrahedral voids (Figure 1.20) are the two most common types of interstitial voids present in the simple spherically close packed (i.e. GCP) metallic structures (fcc and hcp). The former are larger and are surrounded by six atoms which form the corners of a triangular antiprism (octahedron). The latter, which are smaller, are enclosed by four atoms which are tetrahedrally disposed. The primitive unit cell of the hcp structure contains two atoms with coordinates (000) and ( 23 13 21 ). There are thus two atoms associated with each lattice point. If the axial ratio has the ideal value (c/a = 8/31/2 ), then the largest interstices (octahedral) have coordinates ( 13 23 41 ) and ( 13 23 43 ). There are two such interstices per unit cell. The next largest interstices (tetrahedral) occur at (00 38 ), (00 58 ), ( 23 13 81 ) and ( 23 13 87 ), there being

58


a2

a2

120°

a1

a1 Atoms Octahedral voids Tetrahedral voids

Figure 1.20. This figure shows the locations of octahedral and tetrahedral voids in the hcp structure.

four such interstices per unit cell. The region around a tetrahedral void represents the densest packing of equal sized spheres; all topologically close packed structures are characterized by exclusively tetrahedral voids which may be geometrically imperfect because of the differences in the sizes of the component atoms (Sinha 1972). The coordination polyhedron of an atom is defined by the lines joining the centres of atoms in the shell of close neighbours around it. The coordination is twelve-fold in the cases of the fcc and hcp structures and the polyhedra formed by the twelve neighbours assume the shapes of a cubo-octahedron (fcc) and a twinned cubo-octahedron (hcp), respectively. In the case of the TCP structures yet another type of twelve-fold coordination polyhedron, in which all the faces are triangular, becomes important. This polyhedron is the icosahedron which has twenty faces in the shape of equilateral triangles and thirty edges which correspond to nearest neighbour distances. Each of the other two twelve-fold coordination polyhedra mentioned earlier has 24 edges. In the case of the icosahedron, the distance between the central atom and any atom on the polyhedron surface is around 10% smaller than that between the atoms on the surface. The atoms on the icosahedron surface are more close packed than in the fcc or hcp structures; however, because of the five-fold symmetry axis associated with the icosahedron, it is not possible to have a lattice like arrangement made up solely of icosahedra (Sinha 1972). The condition that only tetrahedral interstices may be present in a TCP structure brings in the requirement that besides a number of atoms having an icosahedral environment, certain others with higher coordination polyhedra around them must also be present; thus TCP structures are characterized by some or all of CN 12, CN 14, CN 15 and CN 16 polyhedra (Kasper 1956). All these Kasper polyhedra


59

have exclusively triangular faces and tetrahedral arrangement. The fractions of sites with different coordination numbers change from structure to structure. By way of illustration, let three TCP structures, be considered: A15 (cP8, Cr 3 Si type), C14 (hP12, MgZn2 type) and C15 (cF24, Cu2 Mg type); in the first, 25% of the sites correspond to CN 12 and 75% to CN 14 while in either of the other (Laves phase) structures, 67% of the sites correspond to CN 12 and 33% to CN 16 (Sinha 1972). It has been pointed out (Frank and Kasper 1958, 1959) that there is a significant consequence of there being, in a crystal structure, coordination polyhedra of CN 12, CN 14, CN 15 and CN 16 and exclusively tetrahedral voids and this is that the resulting structure is generally a layered structure. In fact, most of the TCP structures can be regarded as layered structures. The main atomic layers, referred to as primary layers, are tesselated and contain arrays made up of triangles, pentagons and hexagons. The triangular meshes in the primary layers correspond to the nearest neighbour atoms. Besides the primary layers, generally there are secondary layers in which the coordination does not correspond to nearest neighbours (Sinha 1972). The layer stacking has, of course, to be effected in such a manner that only tetrahedral interstices are present. It has been mentioned in the previous section that certain structures are quite frequently encountered in respect of Ti–X and Zr–X intermetallic phases. A brief account will now be presented to illustrate how these structures can be viewed in terms of the layer stacking sequence representation. The two atomic species considered will be designated as X and Z. First, the case of superstructures based on close packed layers stacked in close packing will be taken up. If X and Z are each on a 44 subnet in each close packed layer, then a family of polytypic structures with XZ stoichiometry and a rectangular arrangement of the components in close packed layers is obtained. An example of such a structure is the L1o (tP4, AuCu type) structure. This structure can also be described in terms of stacking alternate 44 layers of X and Z atoms in succession in the [001] direction. One can next consider the case of XZ3 stoichiometry in each close packed layer, with X atoms on a 36 subnet and Z atoms on a 3636 subnet. A family of polytypic structures with a triangular arrangement of X atoms is obtained. The L12 (cP4, AuCu3 type) and D019 (hP8, Ni3 Sn type) structures belong to this family. In the former, the close packed layers, which lie normal to the [111] direction, are stacked in the sequence ABCABC so that all layers are surrounded cubically. In the latter, the close packed layers are arranged in hexagonal ABAB , stacking. The D019 structure is a hexagonally stacked prototype of the L12 structure. It is thus a superstructure of the hcp (hP2) structure in the same way as the L12 structure is of the fcc (cF4) structure (Pearson 1972). Some superstructures based on bcc packing may now be considered. If one considers the XZ stoichiometry, with X atoms on 44 nets and Z atoms also on 44

60


nets, one arrives at the B2 (cP2, CsCl type) structure. In this structure one species occupies the cube corners and the other the body centre, so that alternate layers of the two species occur along directions. The X and Z atoms together form triangular nets parallel to 110 planes with X occupying one of the rectangular 44 subnets resulting from the geometry of the triangles of the 36 net, and Z occupying the other. Next, let the case of the XZ2 stoichiometry be taken up. If one considers 36 close packed layers in bcc [110] stacking with Z atoms on a 63 subnet and X atoms on a larger 36 subnet, one arrives at a family of polytypic structures XZ2 with close packed layers stacked in bcc sequence. The C11b (tI6, MoSi2 type) structure belongs to this family. The C16 (tI12, CuAl2 type) structure can be visualized as being made up of square-triangle, 32 434 nets of Z atoms at z = 0 and z = 21 which are oriented antisymmetrically with respect to each other; the squares in these Z layers which lie over the cell corners and basal face centre, are centred by a 21 44 net of X atoms at z = 41 and z = 43 (Pearson 1972). Even though many structures contain atoms in triangular prismatic coordination, there are some in which this is the main feature of the atomic arrangement. In one class of such structures, 36 and 63 nets of atoms, occupying the same stacking sequence, are stacked alternately in the “paired-layer” sequence (e.g. AaAa). The C32 (hP3, AlB2 type) structure can be regarded as the prototype of this family of structures and one of the simpler structures obtained from this structure is the Bf (oC8, CrB type) structure, which is made up of independent layers of triangular prisms of X atoms parallel to the (010) plane with the prism axes oriented in the [100] direction. The prisms are centred by atoms which form zigzag chains running in the [001] direction (Pearson 1972). An example of structures generated by the stacking of pentagon-triangle nets of atoms is the D88 (hP16, Mn5 Si3 type) structure. In the tetrahedrally close packed A15 (cP8, Cr3 Si type) structure of XZ3 stoichiometry, the X atoms form a bcc array and lines of Z atoms run throughout the structure parallel to the edges of the body centred cell formed by the X atoms. This structure is of the Frank–Kasper type and can be visualized as being formed by the alternate stacking of primary triangle-hexagon 32 62 + 3636 (2:1) layers and secondary 44 layers, with the result that each X atom is surrounded icosahedrally by 12 Z atoms and each Z atom is surrounded by four X atoms and 10 Z atoms in a CN 14 polyhedron with triangular faces. The Laves phase structures have XZ2 stoichiometry and belong to a family of polytypic structures in which three closely spaced 36 nets of atoms are followed by a 3636 kagome net parallel to the (001) plane when the structures are described in terms of a hexagonal cell. The former types of nets are stacked on the same sites as the latter. Alternatively, the Laves phases can be visualised


61

as having Frank–Kasper structures in which pentagon–triangle primary layers of atoms are stacked alternately, parallel to the (110) planes of the hexagonal cell, with secondary 36 triangular layers whose atoms centre the pentagons of the main layers (Pearson 1972). Thus, the C14 (hP12, MgZn2 type) structure is generated by stacking together pairs of primary pentagon–triangle 3535 + 353 (2:3) layers and secondary 36 layers parallel to the (110) plane. The cubic C15 (cF24, MgCu2 type) structure can also be regarded as being built up by stacking consecutively three triangular (36 ) layers and a kagome (3636) layer of atoms which lie in planes normal to the [111] direction in respect of the cubic cell. Each X atom is surrounded by a CN 16 polyhedron of 12 Z and four X atoms while each Z atom is icosahedrally enclosed by six X and six Z atoms. 1.7.3 Derivation of intermetallic phase structures from simple structures The structures of many intermetallics can be regarded as being derived from three simple structures, namely, fcc (A1), bcc (A2) and hcp (A3) structures, which are commonly associated with pure metals and disordered metallic solid solutions. The most common structures exhibited by binary intermetallic phases are listed in Table A1.4 (Ferro and Saccone 1996). Typical intermetallic phase structures derived from the fcc structure include L12 (cP4, AuCu3 type), C15b (cF24, AuBe5 type), L 12 (cP5, Fe3 AlC type), D022 (tI8, TiAl3 type), L1o (tP4, AuCu type), D1a (tI10, Ni4 Mo type), L11 , (hR32, CuPt type) and Pt2 Mo type (oI6) structures (Pitsch and Inden 1991, Sauthoff 1996). Generally these structures are cubic, tetragonal (often with an axial ratio close to unity), rhombohedral or orthorhombic. Examples of intermetallic phases with some of these structures in Ti–X and Zr–X systems are: TiCo3 , TiIr 3 , -TiNi3 , -TiPt3 , TiRh3 , TiZn3 , Zr 3 Al, ZrHg3 , Zr 3 In, ZrIr 3 , ZrPt 3 , ZrRh3 L12 ; ZrNi5 C15b ; TiAl3 , TiGa3 , -ZrIn3 D022 ; TiAl, -TiCu3 , TiGa, TiHg, Ti3 In2 , -TiRh, ZrHg (L1o ); and TiAu4 , -TiCu4 , TiPt8 D1a . Common intermetallic phase crystal structures derived from the bcc structure incldue B2 (cP2, CsCl type), B32 (cF16, NaTl type), D03 (cF16, BiF3 type), and L21 (cF16, Cu2 AlMn type) structures. (Pitsch and Inden 1991, Sauthoff 1996). Generally these structure are cubic. Examples of intermetallic phases with the B2 structure in Ti–X and Zr–X systems are: -TiAu, TiBe, TiCo, TiFe, -TiIr, TiNi, TiOs, -TiPd, -TiPt, -TiRh, TiRu, TiTc, TiZn, S-ZrCo, ZrCu, ZrIr, ZrOs, -ZrPt, -ZrRh, ZrRu and ZrZn. Intermetallics with other bcc-based structures are rare in these systems. Prominent among the intermetallic phase crystal structures derived from the hcp structure are: Bh (hP2, WC type), D019 (hP8, Ni3 Sn type), B19 (oP4, AuCd type), C49 (oC12, ZrSi2 type) and D0a (oP8, -Cu3 Ti type) structures. Generally, these structures are hexagonal or orthorhombic. Examples of intermetallic phases

62


with these structures in Ti–X and Zr–X systems include Zr3 Se2 Bh ; Ti3 Al, Ti3 Ga, Ti3 In, Ti4 Pd, Ti4 Sb, Ti3 Sn, Zr 3 Co, ZrNi3 D019 ; -TiAu, -TiPd, -TiPt (B19); ZrGe2 , ZrSi2 (C49); and -TiCu3 , ZrAu3 D0a . 1.7.4 Intermetallic phases with TCP structures in Ti–X and Zr–X systems It has been mentioned earlier that quite a few of the intermetallic phases occurring in Ti–X and Zr–X systems have topologically closed packed (TCP) structures. These phases are mostly A15 (cP8, Cr 3 Si type) phases or Laves phases (C14, C15 or C36 structures). Examples of such phases are: Ti3 Au, Ti3 Hg, Ti3 Ir, Ti4 Pd (stoichiometric), Ti3 Pt, Ti3 Sb, Zr 3 Au, Zr 3 Hg, Zr 4 Sn, Zr 4 Tl (A15); -TiCr2 , TiFe2 , TiMn2 , TiZn2 ZrAl2 , -ZrCr2 , ZrMn2 , ZrRe2 , ZrRu2 , ZrTc2 (C14); TiBe2 , TiCo2 , -TiCr2 , -ZrCr2 , ZrFe2 , ZrIr 2 , ZrMo2 , ZrV2 , ZrW2 , ZrZn2 (C15); TiCo2 , -TiCr2 and -ZrCr2 (C36, hP24, MgNi2 type). The phase Zr 4 Al3 (hP7) is also a TCP phase the structure of which can be described either in terms of pentagon–triangle primary and 44 secondary nets parallel to the (110) plane, or with hexagon–triangle primary and 36 secondary nets parallel to the (001) plane (Pearson 1972). Some of the Laves phases mentioned above can absorb very significant quantities of hydrogen and, for this reason, are considered for applications as hydrogen storage materials. Reference must be made in this context to the phases ZrV2 , ZrCr 2 , ZrMn2 and TiCr 2 which exhibit high sorption capacities with hydrogen to metal ratio (H/M) values of 1.8, 1.3, 1.2 and 1.2, respectively (Sauthoff 1996). 1.7.5 Phase stability in zirconia-based systems Zirconia (ZrO2 )-based systems are among the most extensively investigated ceramics in so far as phase transformation studies are concerned. Not only do they exhibit interesting phase transformations, but also the properties of these ceramics can be engineered by suitably controlling the stability of different competing phases and by inducing phase transformations in a desired manner. In view of this, zirconia-based systems are pedagogically very appropriate systems for illustrating how phase transformations can be effectively utilized for controlling microstructure and, in turn, properties – mechanical, thermal, electrical and optical – of ceramics. Crystal structures and stability of different phases in zirconia ceramics are briefly described in this section. 1.7.5.1 ZrO2 polymorphs Pure ZrO2 exhibits three polymorphic forms under ambient pressure; these belong, respectively, to monoclinic, tetragonal and cubic crystal systems (Garvie 1970). The crystal structures and lattices parameters of these polymorphs are given in Table 1.18


63

Table 1.18. Crystal structures and lattice parameters of zirconia polymorphs. Phase

Crystal structure

Lattice parameters (nm)

Pearson symbol

Space group

m-ZrO2

mP12

P21 /c

t-ZrO2

tP6

P42 /nmc

c-ZrO2

cF12

¯ Fm3m

a = 0.5156 b = 0.5191 c = 0.5304 = 98 9o a = 0.5094 c = 0.5177 c/a = 1.016 a = 0.5124

(Stevens 1986, Massalski et al. 1992). The occurrence of an orthorhombic form of ZrO2 under high pressures has also been reported (Lenz and Heuer 1982). The monoclinic phase (generally designated as m-ZrO2 ) is stable upto about 1443 K where it transforms to the tetragonal phase (t-ZrO2 ) which is stable upto 2643 K; at still higher temperatures, the cubic phase (c-ZrO2 ) is encountered which is stable upto the melting temperature of 2953 K (Stevens 1986). Among these three phases, the monoclinic phase, has the lowest density. In m-ZrO2 , the Zr 4+ ion has seven-fold coordination with O ions, with a range of Zr–O bond lengths and bond angles. The OII coordination is close to tetrahedral with only one angle (134 3o ) differing significantly from the angle of the tetrahedron (109 5o ) while the OI coordination is triangular. The Zr ions are located in layers parallel to (100) planes, separated by OI and OII ions on either side. The average Zr–OI and Zr–OII distances are 0.207 and 0.221 nm, respectively (Stevens 1986). Figure 1.21 shows a schematic of the idealized ZrO7 polyhedron. Each Zr 4+ ion in t-ZrO2 is surrounded by eight O ions. There is some distortion in this eight-fold coordination due to the fact that while four of the O ions are at a distance of 0.2065 nm, in the form of a flattened tetrahedron, the other four are at a distance of 0.2455 nm in an elongated tetrahedron rotated through 90o (Stevens 1986). The high temperature cubic phase, c-ZrO2 , has the fcc fluorite (CaF2 ) type structure, in which each Zr 4+ ion is coordinated by eight equidistant O ions which are arranged in two equal tetrahedra. A layer of ZrO8 groups in c-ZrO2 is shown in Figure 1.22. 1.7.5.2 Stabilization of high temperature polymorphs An important concept which is often utilized in zirconia ceramics is to “alloy” pure ZrO2 with another suitable oxide to fully or partially stabilize high temperature polymorphs of ZrO2 to lower temperatures.

64


a

a

Zr O

Figure 1.21. Schematic showing the idealised ZrO7 polyhedron pertinent to m-ZrO2 . I I I

II

II II

II

Zr O

Figure 1.22. This figure shows a layer of ZrO8 groups in c-ZrO2 .

The tetragonal to monoclinic transformation, which is martensitic in nature, is accompanied by a large (3–5%) volume expansion which is sufficient to exceed elastic and fracture limits even in relatively small grains of pure ZrO2 and can only be accommodated by cracking. A consequence of this is that the fabrication of large components of pure ZrO2 is not possible due to spontaneous failure on cooling.


65

The addition of cubic stabilizing oxides in appropriate amounts can permit the cubic polymorph to be stable over a wide range of temperatures: even from room temperature to its melting temperature. The oxides that are commonly used to form solid solutions with ZrO2 include MgO (magnesia), CaO (calcia) and RE oxides such as Y2 O3 (yttria) and CeO2 (ceria). These oxides exhibit extensive solid solubility in ZrO2 and are able to form fluorite type phases which are stable over wide ranges of composition and temperature. If the amount of stabilizing oxide added to ZrO2 is insufficient for complete stabilization of the cubic phase, then a partially stabilized zirconia (PSZ) is obtained rather than a fully stabilized form. The PSZ usually comprises a mixture of two or more phases. Both the cubic solid solution and the tetragonal solid solution are present and the latter may transform to the monoclinic solid solution on cooling. It may be mentioned here that the volume expansion associated with the tetragonal to monoclinic transformation may be used to advantage for improving toughness and strength. This aspect will be discussed in a later chapter. A relatively tough, partially stabilized zirconia ceramic, consisting of a dispersion of metastable tetragonal ZrO2 inclusions within large grains of stabilized cubic ZrO2 , can be derived by inducing a stress induced tetragonal to monoclinic transformation. An appraisal of the phase equilibria of zirconia with other oxide systems is very important with regard to the application of zirconia as an engineering ceramic. However, many difficulties are encountered while determining the equilibrium phase diagrams of even the simplest binary zirconia systems. First, the reactions in these systems at relatively low temperatures ( 50 ∼38 33.3 49.5–57 75 ∼83–88 ∼20–∼30 ∼25–33.4 34.9–55.5

P4/mmm I4/mmm P4/nmm I4/mmm P4/nmm Amm2 Pnma I4/m Pmmn P4/mmm ¯ Pm3m P63 /mmc P63 /mmc P63 /mmc 14/mcm P64 /mcm P4/mmm P4/mbm 14/mmm P63 /mcm Fddd ¯ Fm3m I4/mmm P42 /n ¯ Pm3n P4/mmm P63 /mmc P4/mmm ¯ Pm3n ¯ Pm3m ¯ Pm3m P63 /mmc ¯ R3m P42 /mnm ¯ Fm3m I41 /amd ¯ Fd3m ¯ Pm3m P63 /mmc ¯ Pm3m ¯ P 31c ¯ P 3m1 ¯ Fm3m

tP4 tI6 tP4 tI14 tP10 oC12 oP20 tI10 oP8 tP2 cP2 hP12 hP8 hP6 tI32 hP18 tP2 tP32 tI8 hP16 oF24 cF12 tI2 tP6 cP8 tP2 hP8 tP2 cP8 cP2 cP4 hP12 hR53 tP6 cF8 tI12 cF96 cP2 hP16 cP4 hP ∼ 16 hP3 cF8

L6o C11b B11 D1a D0a L1o B2 C14 D019 B82 D8m L1o D022 D88 C54 C1 L 2b A15 L1o D019 L1o A15 B2 L12 C14 C4 B1 Cc B2 D024 L12 B1


75

Table A1.1. (Continued) Phase Ti3 O2 Ti2 O3 Rutile Anatase∗ Brookite∗ TiOs Ti3 P Ti5 P3 TiP TiP2 Ti4 Pb Ti4 Pd Ti2 Pd TiPd (HT) TiPd (LT) Ti2 Pd3 Ti3 Pd5 TiPd2 TiPd3 Ti–Pd phase TiPo Ti3 Pt TiPt (HT) TiPt (LT) Ti3 Pt 5 TiPt 3 Ti–Pt phase TiPt8 Ti5 Re24 Ti2 Rh TiRh (HT) TiRh (LT) Ti3 Rh5 TiRh3 TiRu TiS TiS2 TiS3 Ti4 Sb Ti3 Sb TiSb

Composition (at.% X) ∼40 59.8–60.2 ∼66.7 38–51 25 ∼36–∼ 39 48–50 66.7 ∼20 20 33.3 47–53 47–53 60 62.5 65–67 75 75–84 50 22–29 46–54 46–54 62.5 20.1–40 55–59.2 ∼60–66.7 66.7 93.7 75 66.7 50 33.3

I4/mcm ¯ R3m C2/m P42 /n P63 /mcm Pmm2 Pnma Fddd P63 /mmc P63 /mmc P63 /mcm P63 /mmc Immm ¯ Pm3m ¯ I 43m I4/m ∼P63 /mmc C2/m ¯ P 3m1 P6/mmm Cmcm ¯ Pm3m P63 /mmc ¯ Pm3m I4/mmm

tI12 hP12 mC14 tP32 hP16 oP8 oP8 oF24 hP8 hP6 hP16 hP22 oI44 cP2 cI58 tI18 hP16 mC14 hP3 hP3 oC68 cP4 hP12 cP2 tI6

C16 D88 B27 C54 D019 B82 D88 B2 A12 ∼B81 C6 C32 L12 C14 B2 C11b

∗

Metastable phase; HT: High temperature phase; MT: Medium temperature phase; LT: Low temperature phase.

Table A1.2. Crystal structures of important binary intermetallic phases in Zr–X systems (Massalski et al. 1992). Phase

Composition (at.% X)

Zr 2 Ag ZrAg Zr3 Al Zr 2 Al Zr 5 Al3 Zr 3 Al2

33.3 50 25 33.3 37.5 40

Space group

Pearson symbol

Strukturbericht designation

14/mmm P4/nmm ¯ Pm3m P63 /mmc I4/mcm P42 /mmm

tI6 tP4 cP4 hP6 tI32 tP20

C11b B11 L12 B82 D8m


77

Table A1.2. (Continued) Phase


Zr 4 Al3 Zr 5 Al4 ZrAl Zr 2 Al3 ZrAl2 ZrAl3 Zr 3 Au Zr 2 Au ZrAu2 ZrAu3 ZrAu4 ZrB2 ZrB12 ZrBe∗ ZrBe2 ZrBe5 Zr 2 Be17 ZrBe13 ZrC Zr 2 Cd ZrCd3 Zr 3 Co

42.9 44.4 50 60 66.7 75 25 33.3 66.7 75 80 66.7–68 92.4 50 66.7 83.3 89.5 92.9 33–50 33.3 75 25

Zr–Co phase Zr–Co phase Zr–Co phase Zr–Co phase ZrCr 2 (HT) ZrCr 2 (MT) ZrCr 2 (LT) Zr 2 Cu ZrCu Zr 3 Fe Zr 2 Fe ZrFe2 ZrFe3 Zr 2 Ga Zr 5 Ga3 Zr 3 Ga2 Zr 5 Ga4 ZrGa

33.3 ∼50 >65 to ∼73 79.3 64–69 64–69 64–69 33.3 50 24–26.8 31–33.3 66–72.9 75 33.3 37.5 40 44.4 50

Space group

Pearson symbol


P 6¯ P63 /mcm Cmcm Fdd2 P63 /mmc I4/mmm ¯ Pm3n I4/mmm I4/mmm Pmmn Pnma P6/mmm ¯ Fm3m Cmcm P6/mmm P6/mmm ¯ R3m ¯ Fm3c ¯ Fm3m I4/mmm P4/mmm Cmcm P63 /mmc I4/mcm ¯ Pm3m ¯ Fd3m ¯ Fm3m P63 /mmc P63 /mmc ¯ Fd3m I4/mmm ¯ Pm3m Cmcm I4/mcm ¯ Fd3m ¯ Fm3m I4/mcm P63 /mcm P4/mbm P63 /mcm I41 /amd

hP7 hP18 oC8 oF40 hP12 tI16 cP8 tI6 tI6 oP8 oP20 hP3 cF52 oC8 hP3 hP6 hR19 cF112 cF8 tI6 tP4 oC16 hP8 tI12 cP2 cF24 cF116 hP12 hP24 cF24 tI6 cP2 oC16 tI12 cF24 cF116 tI12 hP16 tP10 hP18 tI16

Bf C14 D023 A15 C11b C11b D0a C32 D2f Bf C32 D2d D23 B1 C11b L6o E1a D019 C16 B2 C15 D8a C14 C36 C15 C11b B2 E1a C16 C15 D8a C16 D88 D5a Bg (Continued)

78


Table A1.2. (Continued) Phase Zr 2 Ga3 Zr 3 Ga5 ZrGa2 ZrGa3 Zr 3 Ge Zr 5 Ge3 Zr 5 Ge4 ZrGe ZrGe2 hydride hydride hydride∗ Zr 3 Hg ZrHg ZrHg3 Zr 3 In Zr 2 In ZrIn ZrIn2 ZrIn3 (HT) ZrIn3 (LT) Zr 3 Ir Zr 2 Ir Zr 5 Ir 3 ZrIr ZrIr 2 ZrIr 3 ZrMn2 ZrMo2 ZrN Zr 2 Ni ZrNi ZrNi3 ZrNi5 ZrO2−x (HT) ZrO2−x (MT) ZrO2−x (LT) ZrOs ZrOs2 Zr 3 P

Composition (at.% X) 60 62.5 66.7 75 25 37.5 44.4 50 66.7 56.7–66.4 63.6 ∼1.0 25 50 75 25 33.3 50 66.7 75 75 25 33.3 37.5 48–53 66.7 70–81 60–79.2 60–67 >40 33.3 50 74–75.5 81.6–85.2 61–66.7 66.5–66.7 66.7 50 >61–∼70 25

Space group

Pearson symbol


Fdd2 Cmcm Cmmm I4/mmm P42 /n P63 /mcm P41 21 2 Pmma Cmcm ¯ Fm3m I4/mmm P42 /n ¯ Pm3n P4/mmm ¯ Pm3m ¯ Pm3m P4/mmm ¯ Fm3m I41 /amd I4/mmm I4/mmm ¯ I 42m I4/mcm P63 /mcm ¯ Pm3m ¯ Fd3m ¯ Pm3m P63 /mmc ¯ Fd3m ¯ Fm3m I4/mcm Cmcm P63 /mmc ¯ F 43m ¯ Fm3m P42 /nmc P21 /c ¯ Pm3m P63 /mmc P42 /n

oF40 oC32 oC12 tI16 tP32 hP16 tP36 oP8 oC12 cF12 tI6 tP6 cP8 tP2 cP4 cP4 tP2 cF4 t124 tI8 tI16 tI32 tI12 hP16 cP2 cF24 cP4 hP12 cF24 cF8 tI12 oC8 hP8 cF24 cF12 tP6 mP12 cP2 hP12 tP32

D023 D88 B27 C49 C1 L 2b A15 L1o L12 L12 L1o A1 D022 D023 C16 D88 B2 C15 L12 C14 C15 B1 C16 Bf D019 C15b C1 C43 B2 C14


79

Table A1.2. (Continued) Phase


ZrP (HT) ZrP (LT) ZrP2 Zr 5 Pb3 Zr 2 Pd ZrPd ZrPd2 ZrPd3 ZrPo Zr 5 Pt3 ZrPt (HT) ZrPt (LT) ZrPt3

50 50 66.7 37.5 33.3 50 66.7 75 50 37.5 50 50 75

Zr 3 Pu ZrPu4 ZrRe2 Zr 5 Re24 Zr 2 Rh ZrRh (HT) ZrRh (LT) Zr 3 Rh5 ZrRh3 ZrRu ZrRu2 Zr 2 S Zr 3 S2 ZrS

26 70–90 66.7 ∼82.8 33.3 50–62 > 50 62.5 72–82 48–52 66–68 33.3 40 50

ZrS2 ZrS3 Zr 3 Sb Zr 5 Sb3 ZrSb2 Zr 2 Se Zr 3 Se2 Zr 2 Se3 ZrSe2 ZrSe3 Zr 3 Si

66.7 75 25 36 66.7 33.3 40 60 64.9–66 75 ∼25

Space group

Pearson symbol


¯ Fm3m P63 /mmc Pnma P63 /mcm I4/mmm ¯ Fm3m I4/mmm P63 /mmc P63 /mmc P63 /mcm ¯ Pm3m Cmcm ¯ Pm3m P63 /mmc P6/mmm P4/ncc P63 /mmc ¯ I 43m I4/mcm ¯ Im3m ¯ Pm3m Cmcm ¯ Pm3m ¯ Pm3m P63 /mmc Pnnm ¯ P 6m2 ¯ Fm3m P4/nmm ¯ P 3m1 P21 /m I 4¯ P63 /mcm Pnnm Pnnm ¯ P 6m2 P63 mc ¯ P 3m1 P21 /m P42 /n I 4¯

cF8 hP8 oP12 hP16 tI6 cF4 tI6 hP16 hP4 hP16 cP2 oC8 cP4 hP16 hP3 tP80 hP12 cI58 tI12 cI2 cP2 oC32 cP4 cP2 hP12 oP36 hP2 cF8 tP4 hP3 mP8 tI32 hP16 oP24 oP36 hP2 hP8 hP3 mP8 tP32 tI38

B1 Bi C23 D88 C11b A1 C11b D024 B81 D88 B2 Bf L12 D024 C32 C14 A12 C16 A2 B2 L12 B2 C14 Bh B1 B11 C6 D0e D88 Bh C6 D0e (Continued)

80


Table A1.2. (Continued) Phase Zr 2 Si Zr 5 Si3 Zr 3 Si2 ZrSi (HT) ZrSi (LT) ZrSi2 Zr 4 Sn Zr 5 Sn3 ZrSn2 ZrTc2 ZrTc6 Zr 3 Te Zr 5 Te4 ZrTe ZrTe2 ZrTe3 Zr 4 Tl U–Zr phase ZrV2 ZrW2 Zr 2 Zn Zr 3 Zn2 ZrZn ZrZn2 ∗

Composition (at.% X) 33.3 37.5 40 50 50 66.7 ∼20 33–∼40 66.7 66.7 85.7 25 44.4 50 55–66.7 75 20 22–37 ∼66.7 ∼66.7 33.3 39.5 50 66.7

Space group

Pearson symbol


I4/mcm P63 /mcm P4/mbm Cmcm Pnma Cmcm ¯ Pm3n P63 /mcm Fddd P63 /mmc ¯ I 43m ¯ R3m I4/m P63 /mmc ¯ P 3m1 P21 /m ¯ Pm3n P6/mmm ¯ Fd3m ¯ Fd3m I4/mmm P42 nm ¯ Pm3m ¯ Fd3m

tI12 hP16 tP10 oC8 oP8 oC12 cP8 hP16 oF24 hP12 cI58 hR12 tI18 hP4 hP3 mP8 cP8 hP3 cF24 cF24 tI16 tP20 cP2 cF24

C16 D88 D5a Bf B27 C49 A15 D88 C54 C14 A12 B81 C6 A15 C32 C15 C15 D023 B2 C15

Metastable phase; HT: High temperature phase; MT: Medium temperature phase; LT: Low temperature phase.

Table A1.3. Nomenclature of crystal structures: strukturbericht designations and corresponding Pearson symbols (Massalski et al. 1992). Strukturbericht designation

Prototype phase

Space group

Pearson symbol

Aa Ab Ac Ad Af Ag Ah Ai

Pa U Np Np HgSn6−10 B Po Po

I4/mmm P42 /mnm Pnma P421 2 P6/mmm P42 /nnm ¯ Pm3m ¯ R3m

tI2 tP30 oP8 tP4 hP1 tP50 cP1 hR1


81

Table A1.3. (Continued) Strukturbericht designation

Prototype phase

Space group

Pearson symbol

Ak Al A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A20 Ba Bb Bc Bd Be Bf Bg Bh Bi Bk Bl Bm B1 B2 B3 B4 B81 B82 B9 B10 B11

Se Se Cu W Mg C (diamond) Sn In As Se C (graphite) Hg Ga Mn Mn I2 Cr 3 Si S P (black) U CoU AgZn CaSi NiSi CdSb CrB MoB WC TiAs BN AsS TiB NaCl CsCl ZnS (sphalerite) ZnS (wurtzite) NiAs Ni2 In HgS PbO CuTi

P21 /c P21 /c ¯ Fm3m ¯ Im3m P63 /mmc ¯ Fd3m I41 /amd I4/mmm ¯ R3m P31 21 P63 /mmc ¯ R3m Cmca ¯ I 43m P41 32 Cmca ¯ Pm3n Fddd Cmca Cmcm 121 3 P 3¯ Cmmc Pbnm Pbca Cmcm I41 /amd ¯ P 6m2 P63 /mmc P63 /mmc P21 /c Pnma ¯ Fm3m ¯ Pm3m ¯ F 43m P63 mc P63 /mmc P63 /mmc P31 21 P4/nmm P4/nmm

mP64 mP32 cF4 cI2 hP2 cF8 tI4 tI2 hR2 hP3 hP4 hR1 oC8 cI58 cP20 oC8 cP8 oF128 oC8 oC4 cI16 hP9 oC8 oP8 oP16 oC8 tI16 hP2 hP8 hP4 mP32 oP8 cF8 cP2 cF8 hP4 hP4 hP6 hP6 tP4 tP4 (Continued)

82



Prototype phase

Space group

Pearson symbol

B13 B16 B17 B18 B19 B20 B26 B27 B29 B31 B32 B34 B35 B37 Ca Cb Cc Ce Cg Ch Ck C1 C1b C2 C3 C4 C6 C7 C8 C9 C10 C11a C11b C12 C14 C15 C15b C16 C18 C19 C21 C22

NiS GeS PtS CuS AuCd FeSi CuO FeB SnS MnP NaTl PdS CoSn SeTl Mg2 Ni CuMg2 ThSi2 PdSn2 ThC2 Cu2 Te LiZn2 CaF2 AgAsMg FeS2 (pyrite) Ag2 O TiO2 (rutile) CdI2 MoS2 SiO2 (high quartz) SiO2 ( crystobalite) SiO2 ( tridymite) CaC2 MoSi2 CaSi2 MgZn2 Cu2 Mg AuBe5 Al2 Cu FeS2 (marcasite) Sm TiO2 (brookite) Fe2 P

¯ R3m Pnma P42 /mmc P63 /mmc Pmma P21 3 C2/c Pnma Pmcn Pnma ¯ Fd3m P42 /m P6/mmm I4/mcm P62 22 Fddd I41 /amd Aba2 C2/c P6/mmm P63 /mmc ¯ Fm3m ¯ F 43m Pa3 ¯ Pn3m P42 /mnm ¯ P 3m1 P63 /mmc P62 22 ¯ Fd3m P63 /mmc I4/mmm I4/mmm ¯ R3m P63 /mmc ¯ Fd3m ¯ F 43m I4/mcm Pnnm ¯ R3m Pbca ¯ P 62m

hR6 oP8 tP4 hP12 oP4 cP8 mC8 oP8 oP8 oP8 cF16 tP16 hP6 tI16 hP18 oF48 tI12 oC24 mC12 hP6 hP3 cF12 cF12 cP12 cP6 tP6 hP3 hP6 hP9 cF24 hP12 tI6 tI6 hR6 hP12 cF24 cF24 tI12 oP6 hR3 oP24 hP9


83


Prototype phase

Space group

Pearson symbol

C23 C28 C32 C33 C34 C35 C36 C37 C38 C40 C42 C43 C44 C46 C49 C54 D0a D0c D0c D0d D0e D02 D03 D09 D011 D017 D018 D019 D020 D021 D022 D023 D024 D1a D1b D1c D1d D1e D1f D1g D13

Co2 Si HgCl2 AlB2 Bi2 Te3 AuTe2 (calaverite) CaCl2 MgNi2 Co2 Si Cu2 Sb CrSi2 SiS2 ZrO2 GeS2 AuTe2 (krennerite) ZrSi2 TiSi2 Cu3 Ti SiU3 Ir 3 Si AsMn3 Ni3 P CoAs3 BiF3 ReO3 Fe3 C BaS3 Na3 As Ni3 Sn Al3 Ni Cu3 P TiAl3 ZrAl3 TiNi3 MoNi4 Al4 U PdSn4 Pb4 Pt B4 Th Mn4 B B4 C Al4 Ba

Pnma Pmnb P6/mmm ¯ R3m C2/m Pnnm P63 /mmc Pbnm P4/nmm P62 22 Ibam P21 /c Fdd2 Pma2 Cmcm Fddd Pmmn I4/mcm I4/mcm Pmmn I 4¯ Im3¯ ¯ Fm3m ¯ Pm3m Pnma P421 m P63 /mmc P63 /mmc Pnma P63 cm I4/mmm I4/mmm P63 /mmc I4/m Imma Aba2 P4/nbm P4/mbm Fddd ¯ R3m I4/mmm

oP12 oP12 hP3 hR5 mC6 oP6 hP24 oP12 tP6 hP9 oI12 mP12 oF72 oP24 oC12 oF24 oP8 tI16 tI16 oP16 tI32 cI32 cF16 cP4 oP16 oP16 hP8 hP8 oP16 hP24 tI8 tI16 hP16 tI10 oI20 oC20 tP10 tP20 oF40 hR15 tI10 (Continued)

84



Prototype phase

Space group

Pearson symbol

D2b D2c D2d D2e D2f D2g D2h D21 D23 D5a D5b D5c D5e D5f D51 D52 D53 D54 D58 D59 D510 D511 D513 D7a D7b D71 D72 D73 D8a D8b D8c D8d D8e D8f D8g D8h D8i D8k D8l D8m D81 D82

Mn12 Th MnU6 CaCu5 BaHg11 UB12 Fe8 N Al6 Mn CaB6 NaZn13 Si2 U3 Pt2 Sn3 Pu2 C3 Ni3 S2 As2 S3 Al2 O3 La2 O3 Mn2 O3 Sb2 O3 (senarmonite) Sb2 S3 Pt2 Zn3 Cr 3 C2 Sb2 O3 (valentinite) Al3 Ni2 Ni3 Sn4 Ta3 B4 Al4 C3 Co3 S4 Th3 P4 Mn23 Th6 ! CrFe Mg2 Zn11 Al9 Co2 Mg32 Al Zn49 Ge7 Ir 3 Ga2 Mg5 W2 B5 Mo2 B5 Th7 S12 Cr 5 B3 W5 Si3 Fe3 Zn10 Cu5 Zn8

I4/mmm I4/mcm P6/mmm ¯ Pm3m ¯ Fm3m I4/mmm Cmcm ¯ Pm3m ¯ Fm3c P4/mbm P63 /mmc ¯ I 43d R32 P21 /c ¯ R3c ¯ P 3m1 Ia3¯ ¯ Fd3m Pnma P42 /nmc Pnma Pccn ¯ P 3m1 C2/m Immm ¯ R3m ¯ Fd3m ¯ I 43d ¯ Fm3m P42 /mnm Pm3¯ P21 /c Im3¯ ¯ Im3m Ibam P63 /mmc ¯ R3m P63 /m I4/mcm I4/mcm ¯ Im3m ¯ I 43m

tI26 tI28 hP6 cP36 cF52 tI18 oC28 cP7 cF112 tP10 hP10 cI40 hR5 mP20 hR10 hP5 cI80 cF80 oP20 tP40 oP20 oP20 hP5 mC14 oI14 hR7 cF56 cI28 cF116 tP30 cP39 mP22 cI162 cI40 oI28 hP14 hR7 hP20 tI32 tI32 cI52 cI52


85


Prototype phase

Space group

Pearson symbol

D83 D84 D85 D86 D88 D89 D810 D811 D101 D102 E01 E07 E1a E1b E11 E21 E3 E9a E9b E9c E9d E9e E9e E94 F5a F01 F51 F56 H11 H24 H26 L 11 L 12 L 2b L 3 L1a L1o L11 L12 L2a L21 L22 L6o

Al4 Cu9 Cr 23 C6 Fe7 W6 Cu15 Si4 Mn5 Si3 Co9 S8 Al8 Cr 5 Al5 Co2 Cr 7 C3 Fe3 Th7 PbFCl FeAsS Al2 CuMg AgAuTe4 CuFeS2 CaTiO3 Al2 CdS4 Al7 Cu2 Fe Al8 FeMg3 Si6 Al9 Mn3 Si AlLi3 N2 CuFe2 S3 Fe3 W3 C Al4 SiC4 FeKS2 NiSbS CrNaS2 CuSbS2 Al2 MgO4 Cu3 VS4 Cu2 FeSnS4 Fe4 N AlFe3 C ThH2 Fe2 N CuPt3 AuCu CuPt AuCu3 CuTi AlCu2 Mn Sb2 Tl7 CuTi3

¯ P 43m ¯ Fm3m ¯ R3m ¯ I 43d P63 /mcm ¯ Fm3m ¯ R3m P63 /mmc Pnma P63 mc P4/nmm P21 /c Cmcm P2/c ¯ I 42d ¯ Pm3m ¯ I4 P4/mnc ¯ P 62m P63 /mmc Ia3¯ Pnma ¯ Fd3m P63 mc C2/c P21 3 ¯ R3m Pnma ¯ Fd3m ¯ P 43m ¯ I 42m ¯ P 43m ¯ Pm3m I4/mmm P63 /mmc ¯ Fm3c P4/mmm ¯ R3m ¯ Pm3m P4/mmm ¯ Fm3m ¯ Im3m P4/mmm

cP52 cF116 hR13 cI76 hP16 cF68 hR26 hP28 oP40 hP20 tP6 mP24 oC16 mP12 tI16 cP5 tI14 tP40 hP18 hP26 cI96 oP24 cF112 hP18 mC16 cP12 hR4 oP16 cF56 cP8 tI6 cP5 cP5 tI6 hP3 cF32 tP2 hR32 cP4 tP2 cF16 cI54 tP4

86


Table A1.4. Binary intermetallic phases: most commonly exhibited structures. Structure type

Number of binary phases exhibiting the structure

Rank


Pearson symbol

Prototype phase

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

A1 A3 B1 A2 B2 L12 C15 D88 C14 C32 Bf D73 D2d D011 B81 C23 C1 L1o A15 C38 B27 C42 B82 C2 D8a

cF4 hP2 cF8 cI2 cP2 cP4 cF24 hP16 hP12 hP3 oC8 cI28 hP6 oP16 hP4 oP12 cF12 tP2 cP8 tP6 oP8 hP38 oI12 hP6 cP12 cF116 hR36

Cu Mg NaCl W CsCl AuCu3 MgCu2 Mn5 Si3 MgZn2 AlB2 CrB Th3 P4 CaCu5 Fe3 C NiAs Co2 Si CaF2 AuCu Cr 3 Si Cu2 Sb FeB Ni17 Th2 CeCu2 Ni2 In FeS2 Mn23 Th6 Be3 Nb

520 362 318 309 307 266 243 177 148 122 120 117 106 101 101 95 87 82 82 74 73 62 61 54 50 49 49

Chapter 2

Classification of Phase Transformations 2.1 Introduction 2.2 Basic Definitions 2.3 Classification Schemes 2.3.1 Classification based on thermodynamics 2.3.2 Classifications based on mechanisms 2.3.3 Classification based on kinetics 2.4 Syncretist Classification 2.5 Mixed Mode Transformations 2.5.1 Clustering and ordering 2.5.2 First-order and second-order ordering 2.5.3 Displacive and diffusional transformations 2.5.4 Kinetic coupling of diffusional and displacive transformations References

89 90 92 93 101 105 105 115 115 116 120 120 122


Chapter 2

Classification of Phase Transformations Symbols G: T: Tc : S: P: H: Cp : : Tm : -phase: -phase: ∗ : e : A : Cij : Ms : V: Vi , Gid : Vd , Gdd :

2.1

and Abbreviations Gibbs free energy ( G = H-TS) Temperature Equilibrium transition temperature Entropy Pressure Enthalpy Specific heat at constant pressure Generalized order parameter Melting temperature hcp phase in Ti and Zr based alloys bcc phase in Ti and Zr based alloys Order parameter corresponding to maximum in free energy Equilibrium order parameter for a first order transition Chemical potential of component, A in the phase, Elastic stiffness modulus (elastic constant) Temperature at which martensite starts forming during quenching Specific volume Velocity and dissipated free energy associated with interface process Velocity and dissipated free energy associated with diffusion process

INTRODUCTION

The study of phase transformations is of interest to metallurgists, geologists, chemists, physicists and indeed to all scientists concerned with the states of aggregation of atoms. Due to the multidisciplinary interest in this subject, a wide variety of nomenclature, sometimes even misleading, has been introduced in the literature for the characterization of different types of phase transformations. It is not uncommon that different sets of terminologies are used in different disciplines for describing essentially similar phase transformations which, in a generalized manner, can be defined as a change in the macrostate of an assembly of interacting atoms or molecules as a result of some variation in the external constraints. The diversity of scientific interest and the complexity of the possible interactions between individual atoms of the assembly naturally lead to many different 89

90


approaches for the study of phase transformations. Physicists primarily focus their attention on higher-order continuous phase transitions in single-component systems such as magnetic, superconducting and superfluid transitions. In contrast, metallurgists and chemists are mainly concerned with phase transformations (which include phase reactions) involving changes in crystal structure, chemical composition and order parameter (both long and shortrange). Phase transformations encountered by geologists, though quite similar to those observed in metallic and ceramic systems, usually occur over much more extended time and length scales under extreme conditions of pressure and temperature. Phase transformations also occur in organic materials such as polymers, biological systems and liquid crystals. Many of the relevant concepts developed for inorganic systems have parallels in organic systems. However, no attempt will be made in this chapter to compare and contrast phase transformations in organic and inorganic systems as the nature of atomic interactions responsible for the transformations is quite different in these two classes of materials. Alloys, intermetallics and ceramics form a group of materials in which phase transformations can be discussed on a common conceptual basis and, therefore, a single classification scheme can be used for appropriately grouping different types of transformations in these systems. As mentioned earlier, Ti- and Zr-based systems, which include alloys, intermetallics and ceramics, exhibit nearly all possible types of phase transformations and, therefore, serve as excellent examples for studies on phase transformations in inorganic materials in general. Phase transformations can be classified on the basis of different criteria, namely, thermodynamic, kinetic and mechanistic (Christian 1965, Roy 1973, Rao and Rao 1978). A comparison of the characteristic features of different types of transformations is presented in this chapter with a view to providing a coarse-brush picture of these in a generalized manner. The chapters which follow will describe these transformations more elaborately, taking illustrative examples from Ti- and Zr-based systems.

2.2

BASIC DEFINITIONS

In order to resolve some of the confusion and controversy which are of a semantic nature a summary of some basic definitions is presented here. A phase is a portion of a system bounded by surfaces and with a distinctive and reproducible structure and composition. Within a single phase, minor fluctuations in structure and/or in composition can occur. One phase can be distinguished from a second phase if at the contacting surface there is a sharp (within one or two atom layers) first-order change in composition and/or structure and hence properties.

Classification of Phase Transformations

91

The two terms phase transformation and phase transition are often used interchangeably. Sometimes a distinction between them is implied but rarely specified. In a paper entitled “A synchretist classification of phase transitions", Rustum Roy (1973) has addressed the controversy which exists in the literature in this regard. Generally the word phase transition is restricted to transitions between two phases which have identical chemical compositions, while the term phase transformation covers a wider spectrum of phenomena which include phase reactions leading to compositional changes. In the metallurgical literature, phase transformations include precipitation of a second phase, , of a different crystal structure and chemical composition from the parent phase ( → + ), and having the same crystal structure but different chemical compositions, eutectoid decomposition ( → + ) and many such processes which, in the chemistry literature, are grouped as phase reactions (Rao and Rao 1978). A more subtle point concerns the meaning of identical chemical composition. The equilibrium point defect concentration may be different in two polymorphs. Though in a strict sense they cannot be considered as identical in chemical composition, transformations between such polymorphs are usually classified as composition-invariant transformations. In considering equilibria between two phases, the requirement of reversibility must be taken into account. Several relationships pertaining to equilibria between two phases can be explained using the free energy versus temperature plot of a single component system (Figure 2.1). The liquid (L) to crystal (A) transition, Equ

ilibr Gla ium ss

A′

tas

tab

oo

rc

le A

TA/B

led liq

Crystal B

pe

Me

Su Cry

stal

uid

Free energy

Tg

A

T L/A

Enantiotropic

Liq

Monotropic

T2

uid

T1

Temperature

Figure 2.1. Free energy versus temperature plots showing phase transformations in a singlecomponent system. The differences between monotropic and enantiotropic transitions and between stable equilibrium and metastable equilibrium transitions are highlighted.

92


L A, at the melting point, TL/A , and the crystal A to crystal B transition, A B, at the transition temperature, TA/B , are stable equilibrium transitions. The transition between a metastable phase A and another metastable phase A , A (metastable) A (metastable), can also be an equilibrium transition and can be grouped along with the former two cases as being enantiotropic, i.e. reversible and governed by classical thermodynamics. In contrast, when we consider transitions which can be represented by vertical lines in this diagram, such as A (metastable) → B (stable) at T1 and Glass → A (stable) at T2 , the reversibility criterion is not met. These irreversible transformations, defined as monotropic transitions, proceed only in one direction and it is not possible to establish an equilibrium between the parent and the product phases. Polymorphic transformations are generally defined as those which involve a structural transition without a change in the chemical composition. Sometimes these transformations are also referred to as congruent processes. There are, however, several examples, such as the transformation of crystalline oxygen to crystalline ozone and transformations of position isomers, which satisfy the aforementioned definition of a polymorphic transition, but cannot even be considered as phase transitions. This is because ozone and oxygen, in the phase rule sense, are two different substances (or components) which survived even the solid → liquid → vapour transitions while preserving their individuality. Similarly each position isomer is an individual component and, therefore, isomeric transitions cannot be considered as phase transitions. In view of this, the definition of polymorphic transformations needs to be restricted to transformations involving phases with different crystal structures which are part of a single component system. In multicomponent metallic alloys and intermetallics, chemical composition-invariant crystallization is a good example in which the parent phase transforms to the product without allowing any partitioning of the constituent elements (or components) between the two phases. In this sense, the system behaves as if it is a single-component system.

2.3

CLASSIFICATION SCHEMES

There are several ways in which phase transformations can be classified, based on thermodynamic, kinetic and mechanistic criteria. A single classification scheme may not be adequate to include all types of transformations encountered in all varieties of materials. In this chapter, an attempt is made to evolve a classification scheme which is applicable to phase transformations in metals, alloys, intermetallics and ceramics.


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2.3.1 Classification based on thermodynamics Ehrenfest (1933) proposed a classification based on the successive differentiation of a thermodynamic potential, usually the Gibbs free energy function, with respect to an external variable such as temperature or pressure. The order of a transformation is then given by the lowest derivative which shows a discontinuity at the transition point. In the generalized sense, for an nth order transition

n G

T n

n−1 G

T n−1

= 0 P

=0

at

T = Tc

(2.1)

where G represents Gibbs free energy and Tc is the equilibrium transition temperature. It is to be noted that the Ehrenfest classification can be used only for equilibrium transitions of a single component system. Substituting n = 1 and 2 in Eq. (2.1), at T = Tc we get for first-order transitions

G = 0

G

T

and for second-order transitions H

G = 0 = − S = −

T P Tc

= − S = P

2 G

T 2

P

H Tc

(2.2)

Cp 1

H = = = 0 Tc

T P Tc (2.3)

A comparison between a first- and a second-order transition can be made in schematic plots of different thermodynamic quantities as functions of temperature (Figure 2.2). First-order transitions are characterized by discontinuous changes in entropy, enthalpy and specific volume. The change in enthalpy corresponds to the evolution of a latent heat of transformation, and the specific heat at the transition temperature, as a consequence, is effectively infinite. In contrast, second-order transitions are characterized by the absence of a latent heat of transformation (as H, S and V do not undergo a discontinuous change at Tc ) and a high specific heat at the transition temperature. There are experimental results which show that in some instances of second-order transitions the specific heat at Tc exhibits infinity rather than a finite discontinuity. A true second-order transition is, therefore, defined as one showing a finite discontinuity in the second derivative of the Gibbs function while a so-called lambda point transition exhibits an infinity. Though the Ehrenfest classification examines the presence of a discontinuity in the nth derivative of the Gibbs function for deciding the order of a transition, in the modern literature transitions with n ≥ 2 are grouped

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Phase Transformations: Titanium and Zirconium Alloys First order

G

Second order

S

G L TM

T

Tc

T

Tc

T

Tc

T

L H

q

S TM

H

T CP

CP

TM

T Temperature

Figure 2.2. Changes in the thermodynamic quantities, free energy, enthalpy and specific heat, at the transition temperatures corresponding to first-order and second-order phase changes.

together as “higher-order transitions” which are characterized by a continuous first derivative, G/ T P = 0, followed by either a discontinuity or infinity for “higher” derivatives. In a multidimensional plot of free energy against temperature, pressure, etc. each phase can be represented by a well-defined surface, as illustrated in Figure 2.3. The equilibrium transformation conditions between two phases are then defined by the intersection of two such surfaces. Moreover the free energy surface for a given phase may be extrapolated into conditions where that phase is not in thermodynamic equilibrium, and the difference in free energy, which is represented by the separation of the free energy surfaces corresponding to the two phases, can be regarded as the driving force for a first-order transformation from one phase to the other. This concept of a metastable phase is not readily applicable to a second-order transformation where it is more appropriate to consider that there is a single continuous free energy surface. Most of the phase transformations encountered in metallic systems are of the first-order type. Ferromagnetic ordering and some chemical ordering processes are examples of higher-order transitions in metallic systems. These transitions can be represented in “mean field” descriptions of cooperative phenomena where the respective order parameters continuously decrease to zero as the temperature


95

β

Free energy

α

e

ur

ss

e Pr

α β Temperature

Figure 2.3. Free energy surfaces for two phases, and , as functions of pressure (P) and temperature (T ). The projection of the line of intersection of the two surfaces on the P–T plane represents the – phase boundary in the P–T phase diagram.

is raised to the transition temperature (Curie temperature or the critical ordering temperature), as shown in Figure 2.4. Any transition which can be described in terms of a continuous change in one or more order parameters can be treated in terms of a generalized Landau equation (Landau and Lifshitz 1969) which states

Tc

η

η Tc

T Second order (a)

T First order (b)

Figure 2.4. Order parameter () versus temperature (T ) plots for (a) second-order and (b) first-order transitions.

96


that close to the critical temperature the free energy difference, G, between finite and zero values of the order parameter, , may be expanded as a power series: G = A2 + B3 + C4 + · · ·

(2.4)

the coefficients A, B, C, etc. being functions of pressure and temperature. The fundamental differences between the first- and higher-order transitions can be explained on the basis of the corresponding Landau plots. For higher-order transitions, the free energy must be an even function of which means that B = 0 (and similarly the coefficients of the odd-powered terms of are zero). Figure 2.5(a) shows the G versus plots for higher-order transitions at different temperatures, both above and below Tc . When T > Tc , the system exhibits a single stable equilibrium at = 0 which corresponds to a positive value of A. As the temperature approaches Tc , the curvature ( 2 G/ 2 at = 0 gradually decreases and as the temperature is lowered below Tc , the curvature as well as the value of A become negative. This essentially means that the system becomes unstable at T = Tc and any infinitesimal fluctuation in the order parameter leads to a lowering First order

Second order

T ≈ Tc Tc > T

η

0

T > Tc Free energy

Free energy

T > Tc

T ≈ Tc Tc > T > Ti

η

0

η = ηc Ti > T

–ve +ve Tc = Ti (a)

Tc >> Ti (b)

Figure 2.5. Free energy as a function of order parameter () for (a) second-order and (b) firstorder transitions. In the case of second-order transitions, the parent phase becomes unstable, ( 2 G/ 2 ) < 0 at = 0 at the transition temperature, Tc , which is the same as the instability temperature, Ti . For some first-order transitions an instability temperature, Ti (which is much lower than the equilibrium transition temperature, Tc ), can be identified where the parent phase becomes unstable at = 0.


97

of the free energy. The negative curvature of the G versus plot also implies that with an increase in the order parameter, the free energy progressively decreases, finally reaching the stable equilibrium positions defined by the minima present in the plots corresponding to T < Tc . The two minima corresponding to the positive and negative values represent two equivalent states associated with antiphase domains of the same ordered structure. Landau plots for a first-order transition are shown in Figure 2.5(b). In this case, the value of B in the Landau expansion (Eq. (2.4)) is not zero. At the transformation temperature, the G versus curve shows two minima, one at = 0 and the other at = c , the two minima being separated by a free energy barrier. At = 0 the system is not unstable as the curvature remains positive at this point at T = Tc . A continuous increase in the order parameter, therefore, will initially raise the free energy which will drop only after the peak of the free energy hill is crossed. Since the system as a whole is not unstable either at Tc or at temperatures close to but below Tc , a gradual transition of the system in a homogeneous manner to the free energy minimum at or near = c is not possible. A phase transition under such a situation can initiate only if localized portions of the system are activated to cross the free energy barrier to reach a point beyond ∗ where can grow further spontaneously. The formation of such localized product phase regions (where has nearly reached the c value) is known as nucleation. The product nuclei remain separated from the parent phase by sharp interfaces and the phase transition proceeds through the growth of these nuclei. The presence of two free energy minima separated by a free energy hill near the equilibrium in the case of a first-order transformation brings out its characteristic features, namely, the coexistence of the parent and the product phases and the discrete nature (involving nucleation and growth) of the transformation. In contrast, all higher-order transitions, by definition, are homogeneous in the sense that the parent and the product phases cannot be distinguished at any stage of the transition and there is no question of having an interface between the two phases. It is interesting to note that a discussion on Landau’s theory, which is strictly concerned only with the equilibrium state, has led us to consider continuous vis-à-vis discrete transformations. Continuous or homogeneous transitions are those in which the parent phase as a whole gradually evolves into the product phase without creating a localized sharp change in the thermodynamic properties and the structure in any part of the system. Such a process can occur only when the system becomes unstable with respect to an infinitesimal fluctuation which leads to the transition and the free energy of the system continuously decreases with the amplification of such a fluctuation. All higher-order transitions, by definition, satisfy the condition of homogeneous/continuous transformation at equilibrium. In contrast, all first-order

98


transitions at equilibrium are discrete transitions which necessarily involve nucleation and growth. When we consider Landau plots for first-order transitions at temperatures far below Tc (Figure 2.5(b)) we notice that at temperatures below the instability temperature, Ti , the curvature of the G versus plot becomes negative at = 0 and the free energy continuously drops with increase in , finally reaching the value corresponding to c . Therefore below Ti a first-order transition can also proceed in a continuous mode. It must be emphasized here that the occurrence of an instability temperature is not universal for all first-order transitions. Only in rather a limited number of cases can first-order transitions be described in terms of a Landau representation. Spinodal clustering, spinodal ordering and displacement ordering processes are some examples in which continuous first-order transitions are encountered in conditions far from equilibrium. Equilibrium phase diagrams showing a miscibility gap correspond to solid solutions which exhibit a clustering tendency. The boundary of the equilibrium two-phase field, 1 + 2 , in the phase diagram (Figure 2.6) is determined by equating the chemical potentials of the two components, A and B, in the two phases, 1 and 2 , in equilibrium at a given temperature, T1 : 1 2 = AB AB

(2.5)

α A

Temperature

α1

X1

B

T1

X 2 α2

Equilibrium solvus Coherent solvus Coherent spinodal

C D

Atom fraction (X)

Figure 2.6. Schematic phase diagram for a clustering system which remains a homogeneous solid solution in region A. A phase separation process occurs by a discrete nucleation and growth mechanism in regions B and C. Spinodal decomposition occurs in region D.


99

The concentrations, X1 and X2 , at which the tie line at T1 intersects the miscibility gap correspond to those of 1 and 2 . A homogeneous solid solution decomposes into an incoherent mixture of 1 and 2 as it is brought from the region A to the region B. Such a decomposition reaction, involving the creation of sharp interfaces between the two phases, is undoubtedly of the first order. A coherent mixture of the two phases can exist in the region C defined by the coherent miscibility gap. The coherent spinodal region D, fully residing within the region C, defines the concentration–temperature field where the phase separation process can initiate by introducing long wavelength (relative to interatomic distances) concentration fluctuations and a continuous amplification of such fluctuations. Though the process is continuous, the transformation is of the first order except at the point where the coherent spinodal touches the coherent solvus, where it may be considered to be of second order. The driving force for the phase separation arising from the negative curvature of the free energy (G)–concentration (X) plot (( 2 G/ X 2 )< 0 in the spinodal region) is opposed by the gradient energy and the coherency strain energy. All these factors and a correction factor for thermal fluctuations determine the wave number for which the amplification rate is the highest. Conceptually a chemical ordering process in which the ordered superlattice can be created only by replacement of atoms in the lattice of the disordered phase can be described in a manner similar to the spinodal clustering process. In the case of continuous ordering, concentration modulations with wavelengths of the order of the interatomic spacing need to be introduced. For an ordering system, the effective gradient energy is negative, and many of the predictions are opposite to those of spinodal decomposition. The amplification factor is negative beyond a critical wavelength and the maximum amplification corresponds to a wavelength equal to a small lattice vector. As shown in Figure 2.7(a) and (b), continuous ordering can be envisaged for both first-order and second-order reactions. Any change in the lattice dimensions due to ordering introduces a third-order term in the Landau equation (Eq. (2.4)) which makes the transformation first order. Continuous ordering in the first-order case requires finite supercooling below the coherent phase boundary. De Fontaine (1975) has also distinguished spinodal ordering from continuous ordering. In the former, the early stages of ordering are characterized by the ordering wave vectors which maximize the amplification factor but the amplification of these does not evolve the equilibrium ordered structure. In true continuous ordering, the equilibrium ordered structure continuously evolves from a low-amplitude quasihomogeneous concentration wave. De Fontaine (1975) has also examined the Landau–Lifshitz symmetry rules for determining whether a specific ordering wave vector qualifies to be a candidate for a second-order transformation.

100

Phase Transformations: Titanium and Zirconium Alloys First order

Second order

Temperature

αd

αd Equilibrium phase boundary

A

A

Coherent phase boundary B

B

αo

C

Coherent instability boundary, T i

D

X

X (a)

(b)

Figure 2.7. Schematic phase diagram for an ordering transition of (a) second order and (b) first order in which continuous ordering occurs under supercooling below the instability temperature, Ti . Region A represents the phase field where the disordered solid solution, d , is stable while Region B corresponds to the stability domain of two-phase mixture, o + d . Region C corresponds to stability domain of ordered o . In the case of a second-order process the d → o transition occurs in a continuous manner below the equilibrium phase boundary. In contrast, continuous ordering is possible in Region D only below the coherent instability boundary, Ti , in some first-order transitions (where the symmetry elements of the ordered structure form a subset of those of the parent disordered structure).

A product phase can evolve from the parent phase through a continuous displacement of the parent lattice. Two types of displacement, namely, a homogeneous lattice deformation and a relative displacement of atoms within unit cells (often called shuffles), can take place, either singly or in combination. The lattice deformation produces a change in volume and external shape and it seems very improbable that this could be accomplished continuously or homogeneously unless the principal lattice strains are very small (within the limits of linear elastic strain). Most martensitic transformations in metals and alloys involve much larger values of lattice strains and are, therefore, not candidates for continuous transformations. However, the possibility of a continuous displacive transformation has to be considered if the lattice deformation is very small. Transformations in some ferroelectric crystals of low symmetry are believed to be of the second order and occur by the progressive development of an instability in a dynamic plane displacement which becomes, at Tc , a static wave extending through the crystal. The approaching soft-mode instability is indicated as a pretransition effect above the transition temperature by a reduction in an appropriate lattice stiffness. Low and reducing values of the shear constant, 21 (C11 − C12 ), are


101

found in the parent phase as Ms is approached from higher temperatures in many noble metal bcc alloys and in fcc In–Tl; but in other martensitic transformations, especially those in steels, there are no anomalies of this kind and, therefore, these transformations are clearly of the first order. The athermal bcc → transformation, which is envisaged as a pure shuffle transition, is another example of a continuous displacive transformation which is a first-order transformation from the consideration of symmetry rules. Though it is possible to accomplish this transformation by a continuous amplification of a displacement wave, the observed fine particle and dual phase ( + ) morphology suggests that -particles form in a quasiperiodic manner through heterophase fluctuations which have the form of ellipsoidal wave packets of displacement wave (Cook 1974). 2.3.2 Classifications based on mechanisms Buerger (1951) has introduced a classification based on mechanisms, namely, reconstructive and displacive transformations. In the metallurgical literature, however, a mechanistic classification groups transformations into (a) nucleation and growth and (b) martensitic types. The introduction of the term nucleation and growth in this context has created considerable confusion, as all first-order transformations including martensitic transformations require the nucleation and the growth steps. In the current literature, usage of such confusing nomenclature is avoided and a mechanistic classification designates the two classes as (a) diffusional and (b) displacive transformations. The former corresponds to the reconstructive transformation in which atom movements from the parent to the product lattice sites occur by random diffusional jumps. This implies that near neighbour bonds are broken at the transformation front and the product structure is reconstructed by placing the incoming atoms at appropriate positions which results in the growth of the product lattice. In contrast, atom movements in a displacive transformation can be accomplished by a homogeneous distortion, shuffling of lattice planes, static displacement waves or a combination of these. All these displacive modes involve cooperative movements of large numbers of atoms in a diffusionless process. Displacive (which includes martensitic) transformations initiate by the formation of nuclei of the product phase, and the growth of these nuclei occurs by the movement of a shear front at a speed that approaches the speed of sound in the material under consideration. In order to differentiate the mechanisms of atom movements across the transformation front, Christian (1965, 1979) has compared the movements involved in diffusional and displacive transformations with civilian and military movements, respectively. In the latter case, if the atoms are labelled in the parent lattice, the coordination between the neighbours can be shown to be essentially retained in

102

Phase Transformations: Titanium and Zirconium Alloys Shear F

F

E

E

D

D

C

C

B

B A

G

H

I

A

J

G

H

I

J

(a) Shear

N′

N M

P′

O′

P O

M′ Q

R

Q′

S

(b)

Figure 2.8. Schematic illustration of lattice correspondence in a two-dimensional lattice. Though the parent and product lattices in both (a) and (b) are identical, the lattice correspondences in the two cases can be distinguished if the dots representing atoms occupying the lattice sites can be labelled. It is through the establishment of the lattice correspondence that the nature of the homogeneous shear and shuffle, if required, can be identified.

the product lattice, though the bond angles undergo changes. This point is illustrated in Figure 2.8(a) which shows how a set of atoms (labelled A, B, C, etc.) decorating the parent lattice changes to the product lattice. The existence of a lattice correspondence implies that a vector in the parent lattice, defined by the sequence of atoms ABCD , becomes a vector in the product lattice with the atoms arranged in the same sequence, although the spacing between them gets altered to match the product lattice dimensions. Such a transformation can be viewed as a homogeneous deformation of the parent lattice (a simple shear in the


103

case of the two-dimensional illustration in Figure 2.8(a), shear directions being shown by arrows). The importance of lattice correspondence in determining the shear can be illustrated by Figure 2.8(b) which shows identical arrays of spots representing the parent and the product structures but with a different lattice correspondence. The atomic rows MNO and MQR are shown in both the parent and the product structures. It is evident from this drawing that the product structure is derived by a combination of a homogeneous deformation and a shuffle. The shear, as indicated by arrows, transforms the rectangle MOPQ in the parent to a parallelogram M O P Q in the product and the atom N is shifted to its new position by a shuffle. The best experimental evidence for the inheritance of the atomic coordination through a displacive transformation is provided by the observation that the chemical order present in the parent structure is fully retained in the product structure. A similar correspondence also exists for crystallographic planes. A relationship of this kind in which straight lines transform to straight lines and planes to planes is described mathematically as an affine transformation. Physically it may be considered as a homogeneous deformation of one lattice into the other. The correspondence associates each vector, plane and unit cell of the parent with a corresponding vector, plane and unit cell of the product. In general, the corresponding lattice vectors and the spacings of corresponding lattice planes are not equal in the two structures, and the angular relation between any pair of lattice vectors in the parent structure is not preserved in the product. It is to be noted that the lattice correspondence does not by itself imply any orientation relation between the phases, since the transformation may involve a rigid body rotation of the product structure with respect to the parent structure. In diffusional transformations, such lattice correspondences are not present. Even in those cases of diffusional transformations in which the chemical compositions of the parent and the daughter phases are identical and a strict orientation relationship exists between them, random jumps of atoms from the parent to the product lattice positions do not permit the lattice correspondence to be preserved. Such composition-invariant diffusional transformations proceed by atomic jumps across the advancing transformation fronts which separate the parent and the product phases. In order to understand the basic difference between displacive and diffusional transformations let us again consider labelling the atoms as A, B, C, D, etc. in the parent lattice and the same set of atoms as A , B , C , D etc. in the product lattice in Figure 2.9(a) and (b), respectively. The transformation front has been shown to advance by a single atomic layer. This schematic drawing shows that the sequence in which these atoms were placed before the transformation is not the same as that in the product lattice. This can happen if each atom breaks the bonds with its neighbours in the parent lattice and shifts to a new position

104


F′

F

E′

E

D′

D C B

C′ B′

A (a) Transformation front F′

F

D′

E

E′

D

C′

C

A′

B

B′ A (b)

Figure 2.9. Schematic diagram of atom movements across the transformation front in a (a) displacive transformation; (b) diffusional transformation.

which corresponds to a lattice point in the product structure. In this manner, the transformation boundary proceeds towards the parent phase, converting the parent to the product phase. The jumps of atoms A, B, C, etc. are random and are not correlated with those of their neighbours, unlike the case of displacive transformations. Since diffusional transformations involve the breaking of bonds between neighbouring atoms and the reconstruction of bonds to form the product phase structure, they are also known as reconstructive transformations.


105

2.3.3 Classification based on kinetics Phase transformations are also grouped in terms of the kinetics of the process. The most important distinguishing kinetic criterion is the requirement of thermal activation. First-order transformations necessarily occur by the nucleation of the product followed by its growth by the propagation of the interface between the parent and the product phases. The movement of the interface can be either thermally activated or athermal. The atom transfer process across the interface is thermally activated in the case of the former while it does not require the assistance of thermal fluctuations in the latter. The kinetic classification, originally introduced by Le Chatelier (Roy 1973), divides phase transformations into two main groups: (a) rapid or nonquenchable and (b) sluggish or quenchable. Transformations belonging to the former class are so fast that the parent phase, which is stable at high temperatures (or high pressures), cannot be retained by a rapid quench to ambient conditions. In contrast, sluggish transitions are slow to the extent that the high-temperature (or highpressure) phase can be retained metastably on quenching. The basic idea behind this classification scheme also centres around the requirement of thermal activation. This classification, however, suffers from the limitation that the experimental ability to rapidly change temperature and pressure is continuously improving and transformations which are grouped as “non-quenchable” today may become “quenchable” tomorrow. A truly displacive transformation occurs through the passage of a glissile interface which is essentially a displacement (or shear) front, the movement of which is not assisted by thermal activation. Such transformations are non-quenchable irrespective of the quenching rate employed.

2.4

SYNCRETIST CLASSIFICATION

The fundamental parameters on the basis of which a phase transformation is classified are thermodynamic, mechanistic and kinetic. A syncretist classification scheme has been introduced by Roy (1973) by taking all these aspects into account. Figure 2.10 shows a three-dimensional matrix with the x-, y- and z-axes representing the mechanistic (structural), thermodynamic and kinetic parameters, respectively. Along the x-axis the two major classes, namely, diffusional (reconstructive) and displacive transformations, are separated by a “mixed” class of transformations which have attributes of both displacive and diffusional transformations. Examples of each of these are available in Ti- and Zr-based systems. Martensitic transformations of the bcc () phase of pure Ti, Zr and of alloys based on these metals have been discussed in detail in Chapter 4 which also deals

106


z

Thermally activated Athermal

Intermediate Rapid

Kinetic

Slow

y

ic

m

na

y od

rm

e Th

≠ t ΔV er rs Fi ord nd a ΔH

0

d ion ith V ixe it w , Δ M rans ect ΔH et ff all pr e sm

=0 d on der ΔV c d r Se o an

ΔH

Displacive

Mechanistic (Structural)

Mixed

Diffusional

x

Figure 2.10. Syncretist classification scheme of phase transformations based on mechanistic (structural), thermodynamic and kinetic criteria.

with martensites in NiTi-based intermetallics and ZrO2 -based ceramics. A host of diffusional transformations such as precipitation, amorphous-to-crystalline phase transformations, massive transformation, eutectoid phase reactions have also been encountered in these systems and they are discussed in Chapters 4 and 7. Displacive transformations can be further divided into different subgroups, depending on whether the transformation is dominated by lattice strains (martensitic transformation) or by shuffles (e.g. omega transition and ferroelectric transitions). The → transition which is frequently observed in several Ti- and Zr-based systems is unique with respect to lattice registry in three dimensions, pretransition effects and transformation product morphology. A detailed account of the -transformation is presented in Chapter 7. The classifications based on thermodynamic criteria, represented along the y-axis of Figure 2.10, divide phase transformations into first-order, higher-order and “mixed” transformations. The distinctions between these classes of transformations are illustrated in Figure 2.11 which depicts the variation of thermodynamic quantities such as specific volume (V ), enthalpy (H) and entropy (S) at and near the transition temperature. In a first-order transition, there is a step change in these quantities at this temperature and there is no need for V , H and S of one phase to show a tendency to gradually approach the value corresponding to the other phase as the transition temperature is approached. A second-order transition is characterized by a gradual change in V , H and S and the absence of a step change in


107

Specific volume Enthalpy Entropy T

T

T

Temperature (a) First order

(b) Mixed

(c) Higher order

Figure 2.11. Changes in specific volume (V ), enthalpy (H) and entropy (S) at and near the transition temperature in (a) second-order and (b) and (c) first-order transitions; (b) shows a “mixed” character, exhibiting pretransition effects as well as step changes at the transition temperature.

these parameters at the transition temperature. The mixed situation is encountered in a large number of transitions such as ferroelectric and ferromagnetic transitions. In such mixed transformations, though a pretransition second-order-like effect is observed, there is a finite discontinuous jump in the value of these thermodynamic quantities at the transition temperature. Considering kinetics as the third variable, phase transformations can be grouped into thermal and athermal classes. All true displacive transformations are athermal which cannot be suppressed by quenching. In contrast, reconstructive or diffusional transformations are invariably thermally activated and, therefore, such transformations are, in principle, suppressible on quenching. The required quenching rate for suppressing a diffusional transformation, however, varies depending on the incubation period involved. Having discussed different schemes of classification of phase transformations in alloys, intermetallics and ceramics, we will now examine how a given phase transformation can be classified on the basis of thermodynamic, kinetic and mechanistic criteria. A classification tree (Figure 2.12) can be constructed by addressing appropriate questions at different levels. The first question to raise is whether the transformation is homogeneous (or continuous) or discrete. Higher-order transitions are continuous while first-order transitions are discrete at the equilibrium transition temperature, Tc . Some firstorder transitions exhibit an instability temperature, Ti (Ti T1 > Mf

Mf Temperature (a)

Temperature

Time

(b)

(c)

Figure 2.13. (a) and (b) Volume fraction transformed as a function of temperature in the case of athermal nucleation of martensite. (c) Increase in volume fraction transformed with time at a constant temperature, T (Ms > T > Mf ), resulting from thermally activated nucleation of martensite.


111

diffusive atom movements occur only across the advancing transformation front. The distances involved in the diffusion process are in the range of the nearest neighbour atomic distances. These transformations are, therefore, designated as short-range diffusional transformations. The partitioning of atomic species occurs between the product phases mainly by interfacial diffusion in cases such as eutectoid decomposition, eutectic crystallization and cellular precipitation, resulting in a two-phase lamellar product. The interlamellar spacing can vary from a few nanometres to a few micrometres. These transformations can be grouped into the category of intermediate-range diffusional transformations. Sometimes classification is made on the basis of the nature of diffusion – whether bulk or interfacial – which dominates in the transformation process. If one draws a comparison between the eutectic and the eutectoid decompositions, one can see that in the former case diffusion is mainly in the liquid phase ahead of the transformation front whereas in the latter case, partitioning of different components occurs primarily at the transformation front. As indicated earlier, displacive transformations are those which can be accomplished by introducing a lattice deformation in the parent lattice. In this class of transformations, a perfect lattice correspondence is maintained and chemical order, if present in the parent phase, is retained in the product phase. A very wide range of transformations are grouped in this class, which covers softmode ferroelectric and ferroelastic transitions in materials such as SrTiO3 and BaTiO3 , the omega transition in Ti and Zr alloys, shear transformations in (bcc) phases in noble metal alloys and martensitic transformations in intermetallics such as nickel aluminides and nickel titanides, cubic-to-tetragonal or cubic-toorthorhombic transitions involving small lattice strains and classical systems of iron-based alloys. Since the characteristics of all of them are not the same, they have been further subdivided into different groups using different criteria for classification. In recent literature, Cohen et al. (1979), Delaye et al. (1982) and Christian (1990) have proposed these classification schemes which are summarized in Table 2.2. It has been mentioned earlier that a displacive transformation involves a homogeneous strain of the parent lattice which is accompanied by atomic shuffles (relative displacements of atoms within unit cells) in some cases. The relative contributions of the lattice strain and the shuffle can be used as important criteria for grouping martensites into two classes, namely, the shuffle dominated and the lattice strain dominated transformations. There are a number of examples of displacive transformations which exhibit pretransition softening of elastic moduli. Amongst these, Ni–Al alloys, containing 30–50% Al, constitute the most widely studied systems; they show weak to moderate first-order character. The fact that the high temperature 2 -phase

112


Table 2.2. Classification of displacive transformations. Criteria Magnitude of shuffle and of homogeneous lattice strain (Cohen et al. 1979)

Presence of precursor mechanical instability (Delaye et al. 1982)

Structural basis (Christian 1990)

Classification Shuffle dominated

Ferroelectric Ferromagnetic Omega

Lattice strain dominated

Martensitic: high strain energy controls morphology and kinetics Quasimartensitic: low strain energy; can occur continuously

No mechanical instability as Ms is approached Strongly first order

Allotropic changes in pure elements Transformations in primary solid solutions

Moderate indications of mechanical instabilities Moderate first order

-phases of noble metal alloys and Ni-based shape memory alloys

Marked mechanical instability Weakly first order, second order

Cubic → tetragonal Cubic → orthorhombic with small lattice strains

Between close packed layer structure

fcc, hcp, 9R, 18R, 4H, etc.; including monolithic and orthorhombic distortions

Between fcc, bcc and derived structure Between bcc, hcp and derived structure Between cubic and tetragonal and slight distortions

(CsCl structure) prepares itself for the transformation as the temperature is lowered towards the martensitic start (Ms ) temperature is well reflected in X-ray and electron scattering as well as in acoustic measurements. The entire 4 0 transverse acoustic phonon branch (corresponding to the shear modulus C in the limit → 0) is unusually low and the energy decreases considerably (though not to zero) at certain wave vectors as the temperature approaches Ms . This partial softening and the evolution of diffuse scattering due to elastic strain along the 0 directions indicate the existence of localized fine-scale displacement patterns. In the Ni625 Al375 alloy, the presence of localized displacements, which are remarkably similar to that required for the formation of the 7R martensite (the


113

stacking sequence of close packed planes being ABAC), has been observed in highresolution electron microscopy images just prior to the transformation (Tanner et al. 1990). As the parent phase approaches the Ms temperature, the density of such microregions deformed by lattice strains increases. The nucleation event in such cases can be viewed as a collapse or growth of microdomains associated with non-uniform lattice strains into a macrodomain of a size larger than the critical size of a nucleus and of homogeneous and nearly appropriate lattice strain. On the question of precursors in martensitic transformations, there are conflicting observations reported in different systems. Martensitic transformations can, therefore, be divided into different subclasses based on the nature and extent of precursor mechanical instability. Strongly first-order martensitic transformations do not show any softening of elastic constants as the system approaches the Ms temperature. In contrast, there are systems where moderate or marked indications of softening of elastic constants are present in the vicinity of the Ms temperature. Generally these “mixed” transformations, exhibiting partial mode softening, are associated with small lattice strains. Displacive-type structural transitions accompanying ferroelectric transitions in BaTiO3 can be cited as examples. These are associated with the displacement of a whole sublattice of ions of one type relative to another sublattice. The perovskite structure with a generalized composition ABO3 consists of a three-dimensionally linked network of BO6 octahedra, with A ions forming AO12 cuboctahedra to fill the spaces between BO6 octahedra. In view of these topological and geometrical constraints, there are only three structural degrees of freedom: (a) displacement of cations A and B from the centres of their cation coordination polyhedra, AO12 and BO6 , respectively; (b) distortions of the anionic polyhedra, coordinating A and B atoms; and (c) tilting of the BO6 octahedra about one, two or three axes. The first of these is the most important for the occurrence of ferroelectricity, since a separation of the centres of positive and negative charges corresponds to an electric dipole moment. Structural phase changes in BaTiO3 with temperature are shown in Figure 2.14. In cubic paraelectric BaTiO3 , both Ba and Ti have zero displacements, with perfectly regular polyhedra of coordinating oxygen ions. The tetragonal, monoclinic (orthorhombic) and rhombohedral forms, which are all ferroelectric, are associated with displacements of the ionic species and distortions of the polyhedra. In tetragonal BaTiO3 , both AO12 and BO6 are elongated along the c-axis, as c/a = 10098. Displacements of 9.68 pm for the Ba2+ ion and 11.50 pm for the Ti4+ ion along the tetragonal axis are responsible for creating a dipole moment with the polarization vector along the same direction. Smaller displacements of the oxygen ions contribute to distortions, with the four oxygen ions in the BO6 octahedron being perpendicular to the tetragonal axis displaced by 3.63 pm, in

114

Phase Transformations: Titanium and Zirconium Alloys Cubic At 130°C; a 1 = a 2 = a 3 = 4.009 Å

Tetragonal

a2

a3

a1

130°C a 1 = a 2 = 3.992 Å

At 0°C

c

a2 a1

c = 4.035 Å Monoclinic a 1 = a 2 = 4.013 Å At c = 3.976 Å –90°C β = 98° 51′

0°C

a3

Rhombohedral

β a1

Orthorhombic c′

α

–90°C a3 a2

α b′

c

a′

a ′ = 5.667 Å At b ′ = 5.681 Å –90°C c ′ = 3.989 Å

α

a1 a 1 = a 2 = a 3 = 3.998 Å At –90°C α = 89° 52.5′

Figure 2.14. Sequence of phase transformations which occur in BaTiO3 at 130, 0 and −90 C. Unit cell dimensions and the orientation of the polarization vector are also indicated.

the opposite direction to the Ti4+ displacement. In the rhombohedral structure, displacements and distortions are correlated. In this case, displacements are parallel to the threefold axis which passes through two opposite triangular faces of the octahedron. The orthorhombic structure, by virtue of its lower symmetry, presents a wider range of polyhedral distortions. As is illustrated in Figure 2.14 the structure of BaTiO3 does not remain stable over the whole temperature range below the first ferroelectric Curie point and it transforms successively into lower symmetry variants, namely, cubic → tetragonal → monoclinic (orthorhombic) → rhombohedral. The vector directions of polarization are also indicated within the unit cells of these structures (Eric Cross 1993). Christian (1990) has grouped martensitic transformations in subcategories based on the structures of the parent and the product phases, as listed in Table 2.2.


2.5

115

MIXED MODE TRANSFORMATIONS

The classification scheme discussed so far makes an attempt to assign a given transformation to a specific category based on thermodynamic, kinetic and mechanistic criteria. It must be emphasized that there exist several phase transformations in real systems which do not fall exclusively in a single category. These transformations, which exhibit characteristics of different classes of transformations, are often called “mixed mode” or “hybrid” transformations (Banerjee 1994). Some of these cases are briefly discussed here for the purpose of illustration. From thermodynamic considerations one can cite cases which show pretransition effects similar to those of second-order transitions and at the same time a sharp discontinuity of thermodynamic functions at the transition temperature. Though these have been designated as “mixed” in Figure 2.11 the sharp discontinuity at the transition temperature makes them first-order transitions as per the thermodynamic definition. Pretransition effects in these often arise due to quasistatic structural fluctuations (modulations in chemical composition or displacement) or modulations in the polarization of electric or magnetic vectors associated with lattice points. The interplay between more than one homogeneous phase transformations can be illustrated by (a) concomitant ordering and clustering processes and (b) sequential operation of spinodal clustering and magnetic ordering. 2.5.1 Clustering and ordering The formation of an ordered intermetallic phase from a supersaturated dilute solid solution often requires concomitant ordering and clustering. The conditions for either simultaneous or sequential operation of clustering and ordering processes have been identified (Kulkarni et al. 1985, Khachaturyan et al. 1988, Soffa and Laughlin 1989) in terms of instabilities associated with the clustering wave vector (k close to 000) and the ordering wave vector (k terminating at one of the special points of the reciprocal space). Let us consider an fcc solid solution which experiences the influences of 100 ordering instability and clustering instability. The following situations can arise and the transformation sequence is accordingly selected: (a) The disordered solid solution is initially unstable with respect to 100 ordering but metastable against clustering. Ordering of the solid solution to an optimum level can introduce a tendency towards phase separation in the optimally ordered structure. (b) The disordered solid solution simultaneously exhibits 000 clustering and 100 ordering tendencies. Both the processes can proceed simultaneously, their relative kinetics determining the rates of progress of the two processes. (c) The peak instability temperature for 000 clustering is higher than that for 100 ordering. In this situation, clustering occurs first, creating solute-rich regions within which the ordering process sets in once the condition of ordering

116


T1 αp

Tricritical point

αp

αf

Paramagnetic

Ferromagnetic P Q

A

αf

R

Composition (%B) (a)

Free energy

Temperature

Tc (X )

A

Composition (%B) (b)

Figure 2.15. (a) A phase diagram showing a two-phase region introduced by a ferromagnetic ordering. (b) Free energy–concentration diagram at T1 showing the introduction of a spinodal clustering region in the ferromagnetic phase.

instability is fulfilled. Such coupled clustering–ordering processes are discussed in Chapter 7 in connection with phase transition sequences in Zr–Al and Cu–Ti alloys. Higher-order transitions like magnetic ordering can also induce a clustering tendency in a solid solution, resulting in the appearance of multicritical points in the phase diagram. Allen and Cahn (1982) have discussed these issues in great detail. Let us consider a binary fcc system of components A and B, where A is ferromagnetic and the Curie temperature, Tc (X), of the A–B solid solution changes with composition in the manner shown in Figure 2.15. In the absence of the magnetic contribution, the system behaves like an ideal solution while with the introduction of the magnetic contribution, the free energy of the -phase is reduced from that corresponding to paramagnetic p to that of ferromagnetic f . At temperatures below the tricritical point, a two-phase region emerges between f and p and two spinodes can also be identified in the free energy–composition plot (Figure 2.15(b)). At a temperature T1 an alloy at the point Q experiences a clustering instability and initially decomposes spinodally to two ferromagnetic phases, the one enriched in the non-magnetic component eventually undergoing a ferromagnetic → paramagnetic disordering. At the points P and R, the alloys are metastable with respect to spinodal clustering and, therefore, the paramagnetic phase, p , in the former and the ferromagnetic phase, f , in the latter can form only by nucleation and growth. 2.5.2 First-order and second-order ordering There are some instances where a system exhibits simultaneously a second-order and a first-order chemical ordering tendency. The ordering process in Ni4 Mo can


117

be cited in this context (Banerjee 1994, Arya et al. 2001). The Ni4 Mo alloy, quenched from the high-temperature disordered (fcc) phase field, exhibits (Spruiell and Stansbury 1965, Ruedl et al. 1968) a short-range ordered (SRO) state characterized by diffraction intensity at 1 21 0 fcc positions and a complete extinction of intensity at 210fcc positions in the reciprocal space. The 1 21 0 reflections do not coincide with the superlattice reflections of the equilibrium Ni4 Mo (D1a ) structure (Figure 2.16). While the SRO state consists of heterospace fluctuations in the form of concentration wave packets of size 2–5 nm, with wave vectors 1 21 0 , the equilibrium D1a structure is associated with wave vectors, 15 420. The two competing superlattice structures, as shown in Figure 2.16(c) and (d), respectively, can be described in terms of 420 planes of all Ni (N) and all Mo (M) atoms in the stacking sequences of MMNNMMNN and MNNNNMNNNN, respectively. While the 1 21 0 ordering fulfils all the symmetry criteria for a second-order transition, the 15 420 ordering is necessarily a first-order transition. In order to examine the relative strengths of the two ordering tendencies, namely, the first order 15 420 and the second order 1 21 0 , the free energy (F ) surfaces, as functions of the respective order parameters, 1 21 0 and 15 420, have been calculated using first principles thermodynamic calculations (Arya et al. 2001). The instability of the system with respect to fluctuations corresponding to the two order parameters can be determined by examining the curvature ( 2 F/ 2 ) of the F versus plots 1 1 at = 0 along the two directions, 1 2 0 and 5 420. With decreasing temperatures, T1 , T2 , T3 and T4 , the following four situations arise, the corresponding free energy surfaces being depicted in Figure 2.17(a)–(d). (1) Tc (D1a ) < Tc (1 21 0) < T1 : positive curvatures for both 1 21 0 and 15 420 ordering, implying stability of the disordered state, = 0. curvature (2) Tc (D1a ) < T2 < Tc (1 21 0): negative curvature for 1 21 0 and positive for 15 420 ordering, implying instability of the system for 1 21 0 ordering and no tendency towards D1a ordering. (3) Ti (D1a ) < T3 < Tc (D1a ), < Tc (1 21 0): negative curvature for 1 21 0 and positive curvature for 15 420 ordering at = 0 but a dip in the free energy with respect to the latter (D1a ordering) near 15 420 = 08. This implies that the system 1 experiences simultaneously tendencies towards 1 2 0 ordering (second order) and D1a ordering (first order). (4) T4 < Ti (D1a ) < Tc (D1a ) < Tc (1 21 0): negative curvatures along both 1 21 0 and 1 420 ordering implying 5 1 that 1the system is unstable with respect to the development of both 1 2 0 and 5 420 In this situation, homoge ordering. neous ordering is feasible for both 1 21 0 and D1a ordering. A mixed state consisting of concentration waves with wave vectors ranging from 1 21 0 to 1 420 is encountered on the path of the ordering process. This mixed state 5

118


(b)

(a)

Period

M

N N M M

N p: 0

1

2

3

4

N2 M 2 (420) -Unit Cell

N2 M2 -Tile

(c) M

M 4 N3 2 1 M

N p: 0 1

2 3

4

5

N4 M

(420)

N4 M

(d)

Figure 2.16. Electron diffraction patterns corresponding to (a) the “short-range ordered” structure characterized by 1 21 0 reflections and complete extinction of 210 reflections and (b) the D1a ordered structure in the Ni–25 at.% Mo alloy. Real lattice descriptions of the fcc-based superstructures in terms of stacking of 420 planes in the [001] projections and static concentration waves are shown in (c) for 1 21 0 ordering and in (d) for D1a -Ni4 Mo ordering with wave vector 15 420. The sequences of Ni (N) and Mo (M) layers of (420) planes and subunit cell clusters are also shown.

Classification of Phase Transformations 6

8

6

T1 > T2 > T3 > T4

4

F ord (K)

4

T1

2

2

T2

T1

0

T2 0

ηc

–2

T3

ηc

T3

–2

–4

T4

T4

–4 0.0

119

0.2

0.4

0.6

0.8

1.0

–6 0.0

0.2

0.4

0.6

0.8

1.0

η /ηmax Figure 2.17. The ordering free energy of the Ni–25 at.% Mo alloy, exhibiting the 1 21 0 and the 1 420 ordering tendencies, plotted as a function of order parameters for the corresponding ordering 5 wave vectors at four different temperatures, T1 , T2 , T3 and T4 , respectively, pertinent to the situations described in the text.

is characterized by diffraction (Figure 2.18) which show a spread of patterns diffracted intensity linking 1 21 0 and 15 420 positions and by the presence of subunit cell clusters (or motifs) representing 1 21 0 and 15 420 ordered structures (as illustrated in Figure 2.16) in lattice resolution images (Figure 2.18).

(a)

(b)

Figure 2.18. Microstructure and diffraction pattern corresponding to mixed 1 21 0 and 15 420 ordering: (a) diffraction pattern showing intensity distribution linking 1 21 0 and 15 420 spots; (b) high-resolution electron micrograph showing motifs of D1a and N2 M2 structures (as schematically illustrated in Fig. 2.16 (c) and (d)).

120


2.5.3 Displacive and diffusional transformations Phase transformations are classified as displacive and diffusional on the basis of the nature of atom movements across the advancing transformation front. One can envisage transformations which occur by a coupling between a displacive and a diffusional mode of atomic movements. The formation of ordered -structures from the disordered parent bcc -phase can be cited as an appropriate example of a mixed diffusive/displacive transformation in which the bcc lattice is transformed into the hexagonal -structure by a periodic displacement of lattice planes while the decoration of the -lattice by different atomic species occurs through diffusional atom movements. These two processes can as well be designated as displacive and replacive ordering, respectively, and the overall process can be viewed as a superimposition of a displacement wave and a concentration wave on the bcc lattice (Banerjee et al. 1997). This mode of transformation (discussed in Chapter 6) is encountered in several bcc Ti and Zr alloys, leading to the formation of a wide variety of ordered -structures. Displacive and diffusional atom movements can also be coupled through kinetic considerations in several cases, one of the best examples being the formation of -hydride precipitates in either the - or the -phase matrix alloys – a topic discussed in detail in Chapter 8. The formation of the -hydride phase from either the - or the -phase involves a shear transformation of the parent lattice accompanied by partitioning of hydrogen atoms (discussed in Chapter 8). The latter process being exceptionally rapid, the displacive lattice shear and hydrogen partitioning can occur nearly concurrently. Hydride formation in Zr- and Ti-based alloys can be compared with bainite formation in steels. 2.5.4 Kinetic coupling of diffusional and displacive transformations Olson et al. (1989) have analysed the kinetics of a transformation process in which the product phase forms with a partial redistribution of the interstitial element during non-equilibrium nucleation and growth. The rate at which the advancing transformation front moves depends both on its intrinsic mobility and on the ease with which the interstitial element diffuses ahead of the moving interface. The intrinsic mobility is related to the process of structural change across the moving interface. The growth, involving partial supersaturation in which local equilibrium is not established at the interface, seems to be an unstable process. This is because a perturbation in composition towards equilibrium would lead to a reduction in free energy which would drive the system to attain a local equilibrium. Stability of the non-equilibrium growth mechanism can, however, be brought about by another process, such as structural transformation across the interface, occurring in series. A displacive structural transition involving movement of a glissile interface and


121

Xl

γ

Gdd

G id

Carbon →

Free energy

α

γ

X

Xα

Xα

Xl Xm

X

Carbon concentration

Distance → (b)

(a)

Figure 2.19. (a) Free energy–concentration plots for ferrite () and austenite () showing the free energy component dissipated in structural change at the interface, Gid , and the free energy component dissipated in diffusive movements of interstitial atoms, Gdd , ahead of the transformation front; (b) shows concentration distribution in the - and -phases.

diffusive movements of interstitial atoms ahead of the interface are, therefore, modelled as coupled processes resulting in the bainitic transformation in steels. Both these processes dissipate the net free energy, G (as indicated in Figure 2.19) which is made up of Gid and Gdd amounts dissipated in the interface process and the diffusion process, respectively. The interfacial velocities can be calculated for the two processes and expressed as Vi = Gid and Vd = Gdd

(2.6)

where and are response functions relating the velocity to the appropriate dissipation. For the process to be kinetically coupled, the interface velocity, V = Vi = Vd . The interface velocity is calculated on the basis of thermally activated motion of dislocations which constitute the ferrite/ austenite glissile interface and is found to be comparable with the diffusion field velocities computed for different levels of interstitial supersaturation. It may be noted that the two types of thermally activated events which are coupled in this treatment operate on widely differing size scales. The manner in which the unit processes of the displacive and the diffusional aspects of the transformation couple at the microscopic level can be understood in terms of the discontinuous nature of the thermally activated interfacial motion. During the “waiting time” prior to a thermally activated event, the solute partitioning in the vicinity of the interface can cause a steady increase in the local interfacial driving force till it reaches a threshold value where the interface is driven to its next position of temporary halt. The movement between these positions occurs through a diffusionless “free glide” motion.

122


This coupled transformation model has been applied to bainitic transformations in steels. The model predicts an increasing interfacial velocity and supersaturation during growth with decreasing transformation temperature, while the nucleation velocity passes through a maximum, giving C-curve kinetics. In this context, it should be mentioned that there is a different view point on the mechanism of the bainitic transformation in which atom transport across the interface is considered to occur through diffusional random jumps. The displacive process, involving coordinated atomic jumps from a parent lattice site to a predestined site in the product lattice, has not been accepted as a requirement for a bainitic transformation in the diffusionist view which has been summarized in an excellent manner by Reynolds et al. (1991).

REFERENCES Allen, S.M. and Cahn, J.W. (1982) Bull. Alloy Phase Diagrams, 3, 287. Arya, A., Banerjee, S., Das, G.P., Dasgupta, I., Saha-Dasgupta, T. and Mookerjee, A. (2001) Acta Mater., 49, 3575. Banerjee, S. (1994) Solid → Solid Phase Transformations (eds W.C. Johnson, J.M. Howe, D.E. Laughlin and W.A. Soffa) TMS, Warrendale, PA, p. 861. Banerjee, S., Tewari, R. and Mukhopadhyay, P. (1997) Prog. Mater., Sci., 42 (1–4), 109. Buerger, M.J. (1951) Phase Transformations in Solids, Wiley, New York, p. 183. Christian, J.W. (1965) Physical properties of martensite and bainite, Special Report 93, Iron and Steel Institute, London, p. 1. Christian, J.W. (1979) Phase Transformations, Vol. 1, Institute of Metallurgists, London, p. 1. Christian, J.W. (1990) Mater. Sci. Eng., A127, 215. Cohen, M., Olson, G.B. and Clapp, P.C. (1979) Proceedings of International Conference on Martensite, MIT Press, Cambridge, MA, p. 1. Cook, H.E. (1974) Acta Metall., 22, 239. de Fontaine, D. (1975) Acta Metall., 23, 553. Delaye, L., Chandrasekaran, M., Andrade, M. and Van Humbeck, J. (1982) Solid– Solid Phase Transformations (eds H.I. Aaronson, D.E. Laughlin, R.F. Sekerka and C.A. Wayman) TMS, Warrendale, PA, p. 1429. Ehrenfest, P. (1933) Proc. Acad. Sci. Amst., 36, 153. Eric Cross, L. (1993) Ferroelectric Ceramics., (eds Nava Setter, Enrico L. Colla and Birkhaeuser Basel), Switzerland. Khachaturyan, A.G., Lindsey, T.F. and Morris, J.W. (1988) Metall. Trans., 19A, 249. Kulkarni, U.D., Banerjee, S. and Krishnan, R. (1985) Mater. Sci. Forum, 3, 111. Landau, L.D. and Lifshitz, E.M. (1969) Statistical Physics, Pergamon Press, Oxford. Olson, G.B., Bhadeshia, H.K.D.H. and Cohen, M. (1989) Acta Metall., 37, 381. Rao, C.N.R. and Rao, K.G. (1978) Phase Transitions in Solids, McGraw Hill, New York.


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Reynolds, W.T., Jr, Aaronson, H.I. and Spanos, G. (1991) Mater. Trans. JIM, 32, 737. Roy, R. (1973) Phase Transitions (ed. L.E. Cross) Pergamon Press, Oxford, p. 13. Ruedl, E., Delavignette, P. and Amelinckx, S. (1968) Phys. Status Solidi, 28, 305. Soffa, W.A. and Laughlin, D.E. (1989) Acta Metall., 37, 3019. Spruiell, J.E. and Stansbury, E.E. (1965) J. Phys. Chem. Solids, 26, 811. Tanner, L.E., Schryvers, D. and Shapiro, S.M. (1990) Mater. Sci. Engi., A127, 205.


Chapter 3

Solidification, Vitrification, Crystallization and Formation of Quasicrystalline and Nanocrystalline Structures

3.1 Introduction 3.2 Solidification 3.2.1 Thermodynamics of solidification 3.2.2 Morphological stability of the liquid/solid interface 3.2.3 Post-solidification transformations 3.2.4 Macrosegregation and microsegregation in castings 3.2.5 Microstructure of weldments of Ti- and Zr-based alloys 3.3 Rapidly Solidified Crystalline Products 3.3.1 Extension of solid solubility 3.3.2 Dispersoid formation in rapidly solidified Ti alloys 3.3.3 Transformations in the solid state 3.4 Amorphous Metallic Alloys 3.4.1 Glass formation 3.4.2 Thermodynamic considerations 3.4.3 Kinetic considerations 3.4.4 Microstructures of partially crystalline alloys 3.4.5 Diffusion 3.4.6 Structural relaxation 3.4.7 Glass transition 3.5 Crystallization 3.5.1 Modes of crystallization 3.5.2 Crystallization in metal–metal glasses 3.5.3 Kinetics of crystallization 3.5.4 Crystallization kinetics in Zr 76 Fe1−x Nix 24 glasses 3.6 Bulk Metallic Glasses 3.7 Solid State Amorphization 3.7.1 Thermodynamics and kinetics 3.7.2 Amorphous phase formation by composition-induced destabilization of crystalline phases 3.7.3 Glass formation in diffusion couples 3.7.4 Amorphization by hydrogen charging 125

128 128 128 135 140 141 145 150 152 153 153 157 157 159 165 171 176 180 182 184 185 187 192 200 205 212 215 220 220 225

3.7.5 Glass formation in mechanically driven systems 3.7.6 Radiation-induced amorphization 3.8 Phase Stability in Thin Film Multilayers 3.9 Quasicrystalline Structures and Related Rational Approximants 3.9.1 Icosahedral phases in Ti-and Zr-based systems References

226 229 237 241 248 252

Chapter 3

Solidification, Vitrification, Crystallization and Formation of Quasicrystalline and Nanocrystalline Structures

List of Symbols Gl/s : Free energy of the liquid/solid phase l/s : Chemical potential of the liquid/solid phase cl/s : Composition of the liquid/solid phase l/s : Activity coefficient for the liquid/solid phase Tl/s : Liquidus/solidus temperature Kl/s : Thermal conductivity in the liquid/solid phase ko : Partition coefficient representing the ratio of the slopes of the liquidus and solidus lines : Wavelength of a perturbation : Amplitude of a perturbation : Rate of growth/decay of a perturbation Gc : Composition gradient of the liquid phase Dl : Diffusion coefficient of the solute tf : Local solidification time Ms : Martensitic start temperature for the phase Tmi : Melting point of the ith component

Hfi : Heat of fusion of the ith component I: Rate of nucleation of a crystalline phase Z: Frequency of the atomic jumps across the interface A∗ : Surface area of the critical nucleus N ∗ : Number density of critical nuclei W ∗ : Work done to form a critical nucleus

Gv : Volume free energy change for the formation of critical nucleus : Induction time f : Fraction of the transformed volume Q: Activation energy of a given process Tp : Temperature at which transformation rate reaches the peak value u: Growth rate ˜ D: Interdiffusion constant

Hv : Enthalpy of vacancy formation 127

128


kB : Boltzmann constant ± : Frequency of ordering + or disordering − jumps Z : Number of nearest neighbor -sites around an -site

3.1

INTRODUCTION

In this chapter, transformations involving liquid, amorphous, nanocrystalline and quasicrystalline phases are discussed. Many of these transformations occur under conditions far removed from equilibrium. With the continuous development of non-equilibrium processing techniques, an increasing number of novel transformation products, some of them possessing exotic properties, have been discovered in the recent past. Ti and Zr alloys have had their due share in this exciting scientific development of production of metastable microstructures by techniques such as liquid and vapour state processing, mechanical attrition, interdiffusion and radiation processing. Phase transformations associated with the liquid, amorphous, quasicrystalline and nanocrystalline states are discussed in this chapter by citing examples taken from Ti- and Zr-based alloys.

3.2

SOLIDIFICATION

Solidification and melting are strong first-order transformations which are of tremendous importance in various technological applications such as ingot casting, foundry casting, single crystal growth and welding. An understanding of the mechanism of solidification is very important for predicting how different parameters such as temperature distribution and cooling rate influence microstructure, alloy partitioning and mechanical properties of cast and fusion welded materials. The objective of this section is to introduce some of the concepts which will be needed for the discussions in later sections dealing with rapid solidification, amorphization and devitrification. 3.2.1 Thermodynamics of solidification As in the case of any other transformation, solidification cannot proceed at equilibrium. Depending on the extent of departure from the equilibrium condition, a hierarchy of different conditions has been identified by Boettinger and Biloni (1996), the solidification rate increasing as the system is driven away from equilibrium. These conditions are listed as follows in the order of increasing departure from equilibrium.

Solidification, Vitrification and Crystallization

129

(1) Full diffusional equilibrium is established globally. Under this condition, there are no gradients of chemical potential and temperature in the system. The compositions of the liquid and the solid phases attain the equilibrium values. The solidification process cannot continue after this condition is attained. (2) Chemical equilibrium is established locally at the liquid/solid interface. The compositions of the liquid and solid phases at the interface are given by the equilibrium phase diagram, any correction due to the interface curvature being taken into account. (3) Local interfacial equilibrium is established between the liquid and a metastable solid phase. Such a situation arises when the equilibrium phase cannot nucleate or grow fast enough to compete with the metastable phase. The compositions of the solid and liquid phases at the interface are given by the pertinent metastable phase diagram. (4) Local equilibrium condition is not established at the liquid/solid interface. Here the temperature and compositions at the interface are not given by either equilibrium or metastable phase diagrams. The condition of local equilibrium, whether stable or metastable, is that the chemical potentials of the components in the liquid and the solid phases are equal across the liquid/solid interface. Condition (4) relates to a situation where the rapidly moving interface does not permit the chemical potentials of the components to equalize across the interface. The rapid growth rates which result under large supercooling can trap the solute into the freezing solid at levels exceeding the equilibrium value for the corresponding liquid composition prevailing at the interface. If one considers only the chemical potential of the solute, it increases upon being incorporated in the freezing solid by a process called solute trapping. However, to make this process thermodynamically possible, a decrease in the chemical potential of the solvent, leading to a net decrease in the free energy, becomes essential (Baker and Cahn 1971). Let us consider the thermodynamics of solidification in a binary system comprising the components A and B, which is represented by the free energy–composition plots, Gl and Gs (Figure 3.1), corresponding to the liquid and solid phases at a temperature which is between the solidus and the liquidus temperatures. In order to determine the composition range of solids which can form from a liquid of composition co , a tangent is drawn to the Gl curve at co . This tangent intersects the Gs curve at two points, cs1 and cs2 . The Gs curve between cs1 and cs2 remains below the tangent, indicating that it is thermodynamically possible to form a solid in this composition range from the liquid of composition, co . At temperatures above the liquidus, tangents to any point on the Gl curve do not intersect the Gs curve and at the liquidus temperature, the tangent to the Gl curve touches the Gs

130

Phase Transformations: Titanium and Zirconium Alloys μsB(c∗n) Tl

Gs

c (To)

A

μ l (co)

P

μsA(c∗s)

Q

μsA(c∗n)

Gl

M

N

cs*

cn* A

μ l (co) cs1

A

cs(eq)

cs2

co

μsB(c∗s)

c l(eq)

Atom fraction of B

B

Figure 3.1. Free energy–concentration plots, Gl and Gs , of the liquid and the solid phases, respectively, at a temperature between liquidus (Tl ) and solidus (Ts ). From thermodynamic considerations, a liquid of composition, co , can form a solid of any composition between cs1 and cs2 .

curve only at one point which gives the equilibrium solid composition, cs eq. If we define the chemical potentials of the components A and B in the liquid and the solid phases at the interface as Al , Bl , As and Bs , respectively, the free energy change during solidification, G, is given by

G = As − Al 1 − cs∗ − Bs − Bl cs∗

(3.1)

where cs∗ is the composition of the solid phase being separated. Considering that the liquid phase is homogeneous in composition, Al and Bl correspond to the liquid composition, co , and are given by the intercepts made by the tangent to Gl at co on the free energy axes corresponding to pure A and pure B, respectively. The free energy change associated with the formation of the solid of composition cs∗ is represented by the drops MN and PQ shown in Figure 3.1 for two different values of cs (cn∗ and cs∗ ). It is to be noted that for cs = cs∗ , solidification involves a lowering of the chemical potentials, A and B , of both the components. In contrast, for cs = cn∗ , A = Al − As < 0 but B = Bl − Bs > 0. Based on the free energy–composition plots for the liquid and the solid phases at a given temperature, the domains of all possible solid compositions, which are allowed to form from thermodynamic considerations, can be presented in an isothermal plot of the solute content of the liquid versus that of the solid at the interface (Figure 3.2). At a given temperature, the liquidus composition gives the

Solid composition at interface, atom fraction of B (cs)


O

Y c (To)

X cs (eq)

Slope 1 ΔG > 0

cs2

B

Δμ > 0

ΔG = 0

Slope 1 Δμ A = 0 Slope k Δμ B = 0

E

Δμ A > 0 Δμ A < 0 ΔμB < 0

131

Equilibrium P

cs1 cl(eq)

Liquid composition at interface, atom fraction of B (c l)

Figure 3.2. The domain OXYEP of all possible solid compositions that can form from various liquid compositions. This can be divided into three distinct regions, the shaded region OEP where chemical potentials of both A and B decrease on solidification, the region above the line OE where solute trapping occurs and the region below the line PE where solvent trapping occurs (after Baker and Cahn 1971).

maximum solute content of the liquid at the interface from which solidification can occur, and in case solidification occurs without any solute partitioning, the solid inherits the composition of the liquid. The condition of partitionless solidification is met at a point where Gl and Gs curves intersect (Figure 3.1(b)) (i.e. where the integral molar free energies of the two phases are equal). The locus of these points defines the To line. The domain of all possible solid compositions that can form from various liquid compositions at a given temperature can be defined by the curve OXYEP. The maximum limit of solute concentration in the solid is given by the point Y , which is fixed by the point of intersection of the cTo line and the line OY (slope = 1) representing identical compositions of the liquid and the solid phases. The point E corresponds to the maximum limit of solute concentration in the liquid, given by the liquidus composition, cl eq, and the equilibrium solidus composition, cs eq. At the boundary of the OXYEP domain, the condition of the change in the integral molar free energy for solidification being zero ( G = 0) is satisfied. As the liquid composition is changed to lower the solute content, a tangent intersects the Gs curve at two points. This situation is illustrated in Figure 3.1 by the tangent at co which intersects Gs at cs1 and cs2 , and solids of any composition between these two limits can solidify from the liquid of co composition. At the composition,

132


cTo , where Gl and Gs intersect, the range of solid compositions spans from zero (point P) to cTo (point Y ). The domain OXYEP can be divided into three regions. In the shaded region OEP, the chemical potentials of both A and B decrease on solidification, while the chemical potential of only one of the components decreases outside this region. The overall free energy change favours solidification in the entire OXYEP domain, but outside the region OEP, one of the components enters the solid phase with an increase in the chemical potential. Such a process is termed solute trapping (in the region OXYE) or solvent trapping (below the line PE, where A > 0 but

B < 0. For a quantitative description of the solute trapping concept, let us consider a simple case of a dilute binary solution for both liquid and solid. Then the chemical potentials are given by Henry’s law for the minor component: Bs = Bs + RT ln s cs

(3.2)

Bl = Bl + RT ln l cl

(3.3)

where Bsl and sl are related constants which depend on the temperature and reference state. Under equilibrium conditions, where cs = cs eq and cl = cl eq, Bs − Bl = Bs − Bl + RT ln

s cs eq =0 l cl eq

(3.4)

The change in chemical potential of the minor component across the solidification front can, therefore, be expressed as

B = RT ln

cs cl eq cs eqcl

(3.5)

In terms of the distribution coefficient, k, at the interface, k = cs /cl ,

B = RT lnk/keq

(3.6)

The straight line OE with a slope keq (in Figure 3.2) represents the equilibrium condition in which the minor component experiences no change in chemical potential. When k > keq, in the region above the line OE, B > 0, which corresponds to solute trapping. For the major component, Raoult’s law holds, and the change in the chemical potential of the component A on solidification can be written as


A = RT ln

133

1 − cs 1 − cl eq 1 − cs eq1 − cl

(3.7)

The straight line PE, of slope 1 − cs eq/1 − cl eq = 1, through the equilibrium composition point cs eq, cl eq in Figure 3.2 represents the equilibrium condition. Solvent trapping occurs below the line PE where A > 0. The G = 0 curve is given by the condition 1 − cs A + cs B = 0

(3.8)

and the curve passes through the points O, X, Y , E and P. Thermodynamics places general restrictions on the composition limits of the solidifying phases, but it does not specify the solid composition under a given supercooling ( T ) and solidification rate (V , the velocity of the solid/liquid interface). Boettinger (1982) has shown the allowable composition ranges of the solid superimposed on phase diagrams of binary systems (Figure 3.3(a) and (b)). The shaded regions in these diagrams indicate thermodynamically allowed solid compositions that may be formed from a liquid of composition co at various temperatures. The To curve gives the highest temperature at which partitionless solidification co = cs can occur. Figure 3.3(b) shows the case where the To curve plunges and partitionless solidification is not permitted for a liquid of composition co . These curves can be used to determine the limit of the extension of solubility obtainable by rapid quenching of the liquid phase. If the To curve plunges to a

co L To

Temperature

Temperature

α

co L To a

Atom fraction of B

Atom fraction of B

(a)

(b)

Figure 3.3. Thermodynamically allowed solid compositions which can form from a liquid of composition co are shown by the shaded regions in two schematic phase diagrams. The To curve gives the highest temperature at which partitionless solidification (co − cs ) can occur. While in (a), partitionless solidification of a liquid of composition co is possible, in (b) where the To curve plunges, partitionless solidification is not permitted for the same liquid composition (after Boettinger 1982).

134


very low temperature, as in Figure 3.3(b), the -phase with solute contents beyond the To curve cannot be formed from the melt. In fact, for phase diagrams with a retrograde solidus, the To curve plunges sharply resulting in a limited extension of solubility. Eutectic systems with plunging To curves are good candidates for easy formation of metallic glasses. This point is elaborated in Sections 3.4.1 and 3.4.2. In contrast, alloys with To curves which are only slightly depressed below the liquidus curves (as shown in Figure 3.3(a)) make good candidates for extension of solubility and are unlikely ones for glass formation. In the analysis of a solidification problem, the condition of local equilibrium is often invoked. This assumption is valid whenever the deviation from equilibrium, expressed as T , cs − cs eq, or cl − cl eq, is small compared to the total temperature and composition ranges pertinent to the solidification process. This is indeed so for most solidification problems that involve rather low velocities of the liquid/solid interface. Let us now consider the steady state plane front condition. We impose a velocity V on the liquid/solid interface and assume that, after an initial transient, the temperature and composition profiles and the position of the interface move with the velocity V . Under this condition, the composition of the solid, cs , must be equal to the overall composition, co , of the alloy. Because cs = co , the process has some aspects of “partitionless” solidification, but it includes the situation in which a liquid layer of a different composition (cl = co ) remains ahead of the advancing solidification front (Figure 3.4). Under steady state conditions, this layer remains unchanged with time and may be hard to detect. It will have a thickness of about Tm

ko < 1

Solid

Liquid

Composition

→

Tl (co)

Initial transient (I) →

Temperature

co

Ts (co)

Initial transient (II) Final transient (III)

Δco co ko co

co/ko

Composition

co ko Δco co c o ko Distance (b)

(a)

Figure 3.4. (a) A schematic phase diagram showing the liquidus and solidus temperatures corresponding to an alloy of composition co . (b) The composition profile of the liquid and the solid in the vicinity of the solidification front under steady state solidification. ko is the equilibrium partitioning ratio.


135

Dl /V where Dl is the diffusivity in the liquid. The thickness of the layer turns out to be less than 1 m when V exceeds 1 cm/s. The possibility of establishing steady state conditions can be examined by redrawing in Figure 3.1 the domains of thermodynamically allowable liquid interface compositions at different temperature – composition regions, defined, respectively, as Regions I, II and III. In Region I, the point B1 , the maximum solute concentration in the solid, remains below co , and therefore, it is not possible to establish the steady state from thermodynamic considerations. This is essentially because G > 0 for all possible values of cl . In Region II, co lies between BII and EII , implying that steady state solidification is possible, provided the composition of the liquid remains within the band marked in the figure. The solid formed remains metastable with respect to partial remelting. There is, however, a diffusional instability for steady state solidification in Region II. A downward fluctuation in the solid composition (cs < co ) will lead to a shift of the liquid composition to the right (cl increasing). Since the solid that is forming is below the average composition co , excess solute is rejected to the liquid, making the liquid further enriched in the solute. This may result either in the break-up of the plane front interface or in a reduction in the interface temperature. Steady state solidification, though thermodynamically possible, is diffusionally unstable. In Region III, steady state growth is not only thermodynamically possible but also stable. If solidification starts at the point EIII , cs > co and the liquid will be depleted of excess solute and the system will spontaneously leave the point EIII and settle on the horizontal line cs = co for partitionless solidification. 3.2.2 Morphological stability of the liquid/solid interface In the previous section, it has been assumed that the solid–liquid interface is microscopically planar. Under this condition, the composition profile induced in the solid varies only in the direction of growth. A planar interface may become unstable to small changes in shape even if the heat flow remains unidirectional. The stability of the liquid–solid interface during solidification is considered here, first for pure metals and then for alloys. In pure metals, solidification can be described in terms of the latent heat being conducted away from the liquid–solid interface, i.e. dT dT = Kl + vLv (3.9) Ks dx s dx l where Ksl are the thermal conductivities of and dT/dxsl are the temperature gradients in the solid and the liquid, respectively, Lv is the latent heat of fusion per unit volume and V is the velocity of the liquid–solid interface. The morphological

136


stability of this interface is governed by the sign of the temperature gradient in the liquid ahead of the transformation front. When the solid grows into a superheated liquid, i.e. for a positive gradient, dT/dxl > 0, the interface remains stable. This can be understood from the following argument. If a small protrusion of solid develops at the plane solidification front due to a local increase in V , the temperature gradient in the liquid ahead of the protrusion will increase, while that in the solid will decrease. Consequently, more heat will be conducted into the protruding solid and less away, resulting in a decrease in the growth rate in this localized region compared to that in the planar region. As a consequence, the protrusion will disappear. The process in which a perturbation of the planar morphology dies down when the latent heat is extracted through the solid is schematically illustrated in Figure 3.5(a).

Solid

Liquid Solid

Liquid

Tm Heat flow

Tm Heat flow T

T

v

v x

x

Solid

Solid

Liquid

Liquid

Heat flow

Heat flow

Isotherms (a)

(b)

Figure 3.5. Temperature distribution during solidification (a) for extraction of heat through the solid and (b) for heat flow into the liquid.


137

During the solidification process, in which the solid grows into a supercooled liquid, the gradient Gl = dT/dx ahead of the interface is negative. If a protrusion forms on such an interface, the negative temperature gradient becomes more negative and, therefore, heat is removed more effectively from the tip of the protrusion than from the surrounding flat regions. A perturbation created on the interface, therefore, tends to grow with time, indicating an inherent instability of the solidification front, as shown in Figure 3.5(b). The instability of the solid–liquid interface is responsible for developing arms on crystals nucleated in a supercooled liquid. These arms grow along crystallographic directions of easy heat transfer and during their growth create secondary and tertiary arms, resulting in a dendritic structure. Dendrites in pure metals are known as thermal dendrites to distinguish them from those forming in alloys primarily due to the constitutional supercooling phenomenon, which is described in the following by considering the case of one-dimensional movement of the solidification front in a binary alloy of composition, co , as shown in the corresponding phase diagram Figure 3.4(a). As in this case the solidification process leads to a partitioning of the solute preferentially towards the liquid phase (partition coefficient, the ratio of the slopes of the liquidus and solidus lines, ko < 1), the liquid ahead of the solidification front becomes solute enriched. After the initial transient, a steady state is established when the liquid in contact with the solidification front attains a composition, co /ko , and the solid/liquid interface reaches the solidus temperature, Ts co . Under this condition, a local equilibrium is established at the solidification front. Tiller et al. (1953) have expressed the composition of the liquid ahead of the solidification front in terms of the solute diffusion coefficient, Dl , in the liquid, the distance, z, from the interface and the velocity, v, of the interface: 1 − ko −vz cl = co 1 + exp (3.10) ko Dl The profile of the solute concentration ahead of the solidification front is shown in Figure 3.6(a) while the liquidus temperature of the solute-enriched region in front of the solidification front is depicted in Figure 3.6(b). The corresponding liquidus temperature for the composition in front of the interface is given by 1 − ko −vz Tl z = Tm + ml co 1 + exp (3.11) ko Dl where ml is the slope of the liquidus line. Figure 3.6(b) also shows three possible profiles of the actual temperature. These are labelled as cases (a), (b) and (c). For the case (a), the actual temperature remains above Tl z, which means constitutional supercooling does not occur ahead of the solid–liquid interface.

138

Phase Transformations: Titanium and Zirconium Alloys a

Slope Gc Solute Temperature gradient, Gs

Solid

c

Temperature

Temperature

Interface

Tl (z) b

Temperature gradient in liquid, Gl

Liquid Distance, z

Distance, z

(b)

(a)

(c)

Figure 3.6. (a) Solute concentration, (b) temperature profile ahead of the solidification front for a system with k0 < 1; z is measured from the solid/liquid interface and (c) early stages of the development of morphological instability of liquid–solid interface as revealed in dendrites of the rapidly solidified Zr 545 Cu20 Al10 Ni8 Ti75 alloy frozen in the amorphous matrix (magnification = 10000X).

Case (b) represents the situation where the actual temperature profile, Tz, is tangent to the Tl z line at z = 0, while for case (c), the Tz line remains below Tl z for some distance ahead of the solid–liquid interface where the condition of constitutional supercooling prevails. The case (b) essentially depicts the limiting condition between the presence and the absence of constitutional supercooling ahead of the solidification front, and this condition can be obtained by equating the slopes of the Tl z and Tz lines at z = 0 which yields Gl ml co ko − 1 ≥ V Dl ko

(3.12)

for constitutional supercooling and the resulting instability in the liquid–solid interface to occur. This criterion for constitutional supercooling, which was obtained by Tiller et al. (1953), serves as a model to understand the major cause of the morphological instability of the solid–liquid interface, but it does not yield any information about the size scale of the modulation developing on the solid–liquid interface. The analysis of morphological stability of the moving solid–liquid interface has been originally reported by Mullins and Sekerka (1963) and many assumptions


139

of the original theory have subsequently been relaxed, as summarized by Sekerka (1986). An outline of the analysis of the morphological stability of the solid–liquid interface can be presented as follows. A perturbation of amplitude, , and wavelength, , is introduced on a flat solid– liquid interface growing in the z-direction. For a two-dimensional z x analysis, the perturbed surface can be represented as z = expt + 2ix/

(3.13)

where is the rate of growth (or decay) of the perturbation. The value of is determined by solving the steady state heat flow and diffusion equations with appropriate boundary conditions for small values of (linear theory). The planar interface is stable if the real part of is negative for all values of and = 0 will define the condition of the stability/instability transition which is given by the following equation: G − ml Gc c +

4 2 Tm =0 2

(3.14)

where is the surface energy and G, the conductivity weighted temperature gradient, is given by G=

Ks Gs + Kl Gl Kl + Ks

(3.15)

Kl and Ks are the conductivities of liquid and solid, respectively. Gc , the composition gradient in the liquid, can be obtained from Eq. (3.10) as Gc =

vco ko − 1 ko Dl

(3.16)

The parameter, c , can usually be set to unity. However, c may deviate significantly from unity under rapid solidification conditions. In general, c is given by c = 1 +

2ko

1 − 2ko − 1 +

4Dl 2 1/2

(3.17)

V

The stability of the solid–liquid interface is determined by the sign of the left hand side of equation (3.14). If it is positive, the interface is stable with respect to the perturbation introduced. The first term G has a stabilizing influence for

140


a positive temperature gradient. For a single component material, this is the only term which is present. Therefore, in such a case, morphological instability can set in only during the growth of the solidification front into a supercooled liquid (negative temperature gradient). The second term in Eq. (3.14) represents the effect of solute diffusion in the liquid and being negative has always a destabilizing influence. The third term, which involves capillarity, has a stabilizing influence for all wavelengths, the minimization of the total surface energy being the motivating factor. This factor becomes more prominent at short wavelengths and, therefore, acts as a balancing force against any reduction in the wavelength of the modulation of the solid–liquid interface. The undulation at the solidification interface in case of a Zr–Al alloy is shown in Figure 3.6(c). 3.2.3 Post-solidification transformations Microstructures of as-solidified alloys based on Ti and Zr are influenced by the solute migration resulting from the solidification process. The first phase to solidify in these alloys in the (bcc) phase which undergoes subsequent solid state phase transformations depending on the local chemical composition (Figure 3.7). This point is explained in Section 3.2.5 by using a hypothetical binary phase diagram for an alloy with a -stabilizing element. Enrichment of the liquid with stabilizing solutes causes a local depression of the Ms temperature. During

Figure 3.7. Bright-field microstructure showing martensitic structure cutting across the cellular boundaries. The martensite has formed from the phase which was the first phase to solidify from the liquid.


141

Table 3.1. Sequences of transformations during solidification processing in Ti- and Zr-based alloys. Alloy system

Sequence of phase transformations

Zr (with Si and O impurities) Zr – 1 at.% Nb Zr – 8.5 at.% Nb Zr – 27 at.% Al Ti – 50 at.% Al Zr3 Al–Nb

L → directly L L L L L

→ → → → → → + → Zr 2 AlB82 + → → 2 → 2 + → → + → + Zr 5 Al3 → Zr 2 Al

post-solidification cooling, some regions encounter a martensitic ( → ) or a Widmanstatten ( → ) transformation, depending on the prevailing cooling rate. Regions which are enriched in the -stabilizing solute beyond a certain level either retain the phase fully or transform into a + structure. The martensitic and the retained (with or without dispersion) structure is superimposed on the dendritic or cellular structure in the final microstructure, with the local chemical composition determining the nature of the post-solidification transformation. The solute enrichment process in the interdendritic regions can also induce other phase reactions such as the formation of an ordered intermetallic phase, → Zr 2 Al B82 , or a peritectoid reaction, + Zr 5 Al3 → Zr 2 Al, or → 2 → 2 + (as in TiAlbased alloys). Transformation sequences during solidification processing have been determined in some limited studies on Ti- and Zr-based alloys. Table 3.1 summarizes the results in a few representative cases. 3.2.4 Macrosegregation and microsegregation in castings The solidification theory discussed in the preceding section is utilized in assessing the extent of macro- and microsegregation in ingots and other castings. Macrosegregation causes non-uniformity in alloy composition that occurs over large distances, while microsegregation is over distances comparable to the dendrite arm spacing. The extent of microsegregation is expressed in terms of the segregation ratio (= local maximum solute concentration/local minimum solute concentration) or the volume fraction of the non-equilibrium secondary phase which forms as a result of segregation. As the solidification front grows into the liquid metal pool, the dendrite arms isolate the liquid into microscopic pools in the mushy zone (Figure 3.8). The region between two adjacent dendrite arms can be taken as a characteristic volume since the dendrite spacing, d, in an alloy under a given cooling condition remains quite uniform. By applying the equilibrium partition ratio, ko , the interface composition

142

Phase Transformations: Titanium and Zirconium Alloys Solid + liquid (mushy zone)

Solid

Liquid

λ = 0 (Dendrite spine) Solid

λ = λ i (Solid–liquid interface)

λ Liquid

λ = d /2 (Midpoint between two dendrites)

xR

xr

Figure 3.8. Schematic diagram showing dendritic areas of the solid phase growing into the mushy zone.

of the solid and the liquid phases can be estimated to be ko co and ko cl where co is the average composition and cl is the liquidus composition at the non-equilibrium solidus, as shown in Figure 3.9. The redistribution of the solute within the solid dendrite by diffusion can be computed from the material balance equation cl∗ − cs∗ dfs =

fs 0

Ds

2 cs df dt + 1 − fs dcl∗ x2 s

(3.18)

where cl∗ and cs∗ are the compositions at the interface of the liquid and the solid, respectively, fs is the weight fraction solid within the volume element and Ds is the diffusion coefficient of the solute in the base metal. The solute redistribution due to diffusion can be computed by substituting values for diffusion coefficient, dendrite arm spacing, solidification time and by employing numerical techniques described in literature (Brody and Flemings 1966). An estimate of the extent of microsegregation can be made by evaluating the parameter a=

4Ds tf d2

(3.19)

where tf is the local solidification time which is defined as the difference in time between the passing of the liquidus and solidus isotherms for a given point in the ingot. For a 1, diffusion in the solid is negligible and microsegregation is maximum, whereas for a 1 microsegregation is negligible. Since d = Ctfn 03 < n < 05 microsegregation does not change with the cooling rate when n = 05.

(3.20)

Temperature


143

Tl

T s′ c ok o

c ′I

co Composition

Solid

Solid + liquid

Liquid

Temperature

Tl

T s′

cR

xT

Liquid composition

Distance

c ′I co

cR

xT Distance

Figure 3.9. Temperature–composition diagram showing interface composition of the solid (ko co ) and the liquid phases (ko cl ) where co is the average composition and cl is the liquidus composition at the non-equilibrium solidus. The liquid composition profile ahead of the solidification front is also shown.

Brody and David (1970) have compared the microsegregation parameters of several binary Ti alloys. It is noted that the extent of microsegregation increases with an increase in the freezing range and with a steep slope of the liquidus. Ternary and more complex alloys are expected to show more extensive microsegregation whenever the addition of alloying elements results in a lowering of the freezing range. Macrosegregation arises out of the bulk movement of solute rich liquid (for ko 1.0 >2.0 >0.5 >1.5

RSP technique

Splat Splat Splat EBSQ Laser EBSQ Splat


153

3.3.2 Dispersoid formation in rapidly solidified Ti alloys Attempts to disperse hard refractory compounds in Ti alloys have not been quite successful in the ingot metallurgy of these alloys. This is due to the fact that stable compounds segregate to a very significant extent during solidification at the conventional ingot cooling rates. Rapid solidification processing has, therefore, been employed in several Ti alloys with the aim of producing a finely dispersed structure. Rapidly solidified Ti alloys with additions of (a) Er or (b) Ce and S have shown very fine (50–100 nm) dispersions of Er2 O3 , CeS and Ce4 O4 S3 . These dispersed phases have shown excellent resistance against coarsening at temperatures as high as 1223 K. Compounds of Ti with the metalloid elements C, B, Si and Ge have high melting points and good chemical stability. These compounds are, however, less stable in the Ti alloy matrix than the rare earth oxides and sulphides. Rapidly solidified Ti alloys containing C show a distribution of fine spherical TiC precipitates while those containing B produce a fine dispersion of needle-shaped TiB precipitates. Ti alloys with Si and Ge, in the rapidly solidified condition, contain fine dispersions of Ti5 (Si,Ge)3 . Both carbides and silicides coarsen rapidly at 973 K while TiB remains stable upto about 1073 K. Rapid solidification processing is also applied for alloys in which eutectoid decomposition takes place. The cooling rate required for solidification without any significant segregation is lower in the case of eutectoid-forming alloys than in rare earth or metalloid-containing alloys. A higher volume fraction of the second phase makes the eutectoid-forming alloys (containing one or more of the alloying elements Cr, Mn, Fe, Ni, W and Cu) suitable candidates for high strength alloys which can retain their strength up to intermediate temperatures. The upper temperature limits of the stability of intermetallic compounds present in eutectoid alloys restrict their high temperature applications. 3.3.3 Transformations in the solid state The influence of the rapid solidification treatment on the subsequent phase transformations in the solid state has not been studied extensively. Inokuti and Cantor (1979) have reported refinement of the martensitic structure in rapidly solidified Fe based alloys. The refinement is attributed to the small size of austenite grains forming from the liquid phase which, in turn, limits the size of the martensite plates. Banerjee and Cantor (1979) have reported the microstructure produced in unalloyed Zr and Zr–Nb alloys by rapid quenching from the liquid state. The possibility of the formation of the -phase directly from the liquid phase, skipping the intermediate equilibrium -phase, has been examined. The thermodynamic feasibility

154


–100

L (1700 K)

↓

Tα

α (Zr)

β (Zr)

L (Zr)

–300

–500

α (Zr) Tα

β (1135 K)

1000

β (Zr)

L (Zr)

Tβ

L (2125 K)

↓

–400

↓

G (kJ/mol)

–200

2000 3000 Temperature (K)

4000

Figure 3.14. Schematic free energy versus temperature plots for the liquid, the - and the -phases for pure Zr. The temperatures T/l and T /l correspond to those at which the liquid/ and the liquid/ equilibrium are established.

of such a process is illustrated schematically in Figure 3.14. The evidence of a direct L → transformation has been recorded in unalloyed Zr samples contaminated with Si and O. The cellular structure of the -phase with Si-enriched intercellular region (Figure 3.15(a)) observed in rapidly solidified samples points to the fact that the -phase cells originated directly from the liquid. The extent of supercooling required for making such a direct L → transformation possible can be reduced by alloying elements which enhance the relative stability of the -phase in comparison with the -phase. Since both Si and O are strong

-stabilizers, contamination of these elements in Zr is expected to raise the Tl/ temperature and thereby reduce the extent of supercooling required for the direct solidification. In alloys containing a -stabilizing element, two types of displacive transformations, namely the → martensitic and the → transformations, are encountered. Which of these processes is selected by a given alloy is determined by the alloy composition. As has been explained in Chapter 1, the → transformation operates in compositions where Ms is higher than Ms and vice versa. The transition from the structure to the + structure is noticed in Zr–Nb


(a)

155

(b)

(c)

Figure 3.15. Microstructures developed in rapidly solidified Zr alloys undergoing post-solidification phase transformation: (a) cellular morphology of the -phase which formed directly from the liquid. Intercell boundary regions are richer in Si and O. (b) Propagation of -martensite laths across the boundaries of -cells which formed as a solidification product. (c) + microstructure in Zr-5.5 wt% Nb alloy.

samples quenched from the -phase field (solid state quenched) when the Nb level exceeds ∼7 at.%, the composition at which Ms line intersects the Ms line (Figure 1.18). Banerjee and Cantor (1979) have shown that in rapidly solidified Zr–Nb samples, this transition is shifted to a lower (∼5.5%) level of Nb. This preference for → transformation over the → martensitic transformation under rapid solidification is attributed to the retention of a higher concentration of vacancies which are known to stabilize the -like defects prior to the transformation (Kuan and Sass 1976). Figure 3.15(b) and (c) shows the martensitic structure and the + structure, respectively, in rapidly solidified Zr–Nb alloys. It may be noted that the laths in (b) are cutting across the intercell boundaries, implying that the concentration profiles across the cell boundaries are such that Ms remains above the quenching temperature all along the growth path of martensite laths.

156


Figure 3.16. Dark-field micrograph of Zr-27 at.% Al alloy showing cellular structure leading to Al enrichment at cell boundaries where higher density of Zr2 Al particles are seen.

In a study on a rapidly solidified Zr-27 at.% Al alloy, Banerjee and Cahn (1983) investigated the sequence of transformation events which led to the formation of the ordered Zr2 Al phase (B82 structure) in this alloy. The analysis of the morphological features (Figure 3.16) of the transformation products led to the conclusion that the following sequence of transformation events occurred: (1) Formation of a supersaturated (bcc)-phase mainly by a partitionless solidification process. The limited alloy partitioning in some localized areas led to the formation of a cellular structure where the cell boundaries were decorated by a higher number density of Zr2 Al particles as shown in Figure 3.16. (2) Spinodal decomposition of the supersaturated -phase (Zr–Al) during continuous cooling subsequent to solidification. This process resulted in the formation of a compositionally modulated structure with modulations along the elastically soft 100 directions which, in turn, produced Al-rich cuboids of about 20 nm size. (3) A combined chemical and displacement ordering within the Al-rich cuboids resulting in the formation of the Zr2 Al structure – details of which are discussed in Chapter 6. (4) → transformation in the Al-depleted regions in the intervening space between the cuboids.


3.4

157

AMORPHOUS METALLIC ALLOYS

Amorphous metallic alloys or metallic glasses have emerged as a new class of engineering materials after vitrification of metallic alloys by using the technique of ultrarapid quenching of molten alloys has become possible. These materials, which do not have any long-range crystalline order but retain metallic bonding, exhibit several interesting properties emanating from their unique structure which is isotropic and homogeneous in the microscopic scale. Extremely high hardness and tensile strength, exceptionally good corrosion resistance and very low magnetic losses in some soft magnetic materials are some of the attractive properties associated with amorphous metallic alloys. There are three technologically important classes of amorphous alloys, namely (a) the metal–metalloid alloys, such as Fe–B, Fe-Ni-P-B and Pd–Si, (b) the rare earth–transition metal alloys, such as La–Ni and Gd–Fe, and (c) the alloys made up of a combination of early and late transition metals such as Ti–Cu, Zr–Cu, Zr–Ni and Nb–Ni. With the availability of metallic glasses, several important issues concerning the stability of these metastable phases and the kinetics of their thermal decomposition have attracted the attention of the phase transformation research community. It will be shown in this chapter that the formation and the decomposition of Ti- and Zr-based metallic glasses offer some unique opportunities for studying several aspects of phase stabilities and transformations in metallic glasses. These include glass-forming abilities (GFAs), diffusion mechanisms, modes and kinetics of crystallization and formation of bulk metallic glasses. A major advantage of studying these systems is that a number of binary metal–metal alloys based on Ti and Zr are amenable to easy glass formation. 3.4.1 Glass formation The three-dimensional lattice arrangement of atoms in a crystalline solid is destroyed as it melts. In the liquid state, the long-range order both translational and orientational of the crystalline solid is not retained as the atoms vibrate about positions which are rapidly and constantly interdiffusing. Melting being a strongly first-order transition, thermodynamic quantities such as specific volume, enthalpy and entropy undergo a discontinuous change at the melting temperature as a crystalline solid transforms into a liquid. At temperatures above the melting point, a liquid is in a state of internal equilibrium and its structure and properties are independent of its thermal history. A low viscosity, which is essentially the inability to resist a shear stress, characterizes the liquid state. On cooling, a liquid transforms into a crystalline solid under the equilibrium cooling condition. Vitrification of a liquid is possible only when the liquid is cooled at a rate sufficiently rapid to escape a significant degree of crystallization so that the “disordered” atomic

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configuration of the liquid state is frozen in. Glass formation is easy in a number of non-metallic systems such as silicates and organic polymers. The nature of bonding in these systems places severe limits on the rate at which crystalline order can be established during cooling. Thus the melt solidifies into a glass even at low cooling rates (often less than 10−2 K/s). Metallic melts, in contrast, have non-directional bonding which allows a very rapid rearrangement of atoms into the crystalline state. Hence very high cooling rates need to be imposed for forming metallic glasses by avoiding crystal formation. Cooling rates exceeding 105 K/s are necessary for the formation of metallic glasses in several binary and ternary alloy systems based on Ti and Zr. In recent years, several Zr-based alloys with a number of components have been found to be amenable to vitrification at substantially slower cooling rates (Inoue 1998). This has opened up the possibility of obtaining metallic glasses in the bulk form. The formation and properties of bulk metallic glasses are discussed in Section 3.6. The process of vitrification of a liquid under non-equilibrium cooling can be compared with the equilibrium crystallization process in plots of viscosity, , and thermodynamic quantities such as specific volume, V , and specific heat, Cp , against temperature (Figure 3.17). While the liquid to crystal transformation is accompanied by a step change in these properties, a progressive change in viscosity and enthalpy precedes the vitrification event as the liquid is undercooled below the equilibrium melting temperature, Tm . It is evident from these plots that vitrification is possible only if the equilibrium crystallization process is avoided. The cooling rate, therefore, needs to be sufficiently high so that insufficient time for nucleation and/or growth does not permit the formation of the crystalline phase to a detectable level. Although the driving force for nucleation continuously increases with the extent of undercooling, the rapid increase in viscosity is responsible for decreasing atomic mobility and thereby effecting the kinetic suppression of crystallization. Eventually, the atomic configuration of the liquid becomes homogeneously frozen at the glass transition temperature, Tg . This structural freezing to the amorphous state is, by convention, considered to occur when the viscosity reaches a value of 1013 poise. Since the atomic configuration of the amorphous state does not correspond to a unique equilibrium structure, Tg and the glass structure are both cooling rate dependent, variations in the latter resulting in glasses with different states of structural relaxation. Figure 3.17(a) shows the glass transition temperatures, Tg1 and Tg2 , for two glasses, G1 and G2 , forming under different rates of quenching. The GFAs of different metallic systems have been assessed in terms of both relative thermodynamic stabilities of the amorphous and the equilibrium and metastable crystalline phases which compete to form during cooling and kinetic factors which determine the critical cooling rate necessary for avoiding the crystallization process. The thermodynamic and kinetic criteria for glass formation

Solidification, Vitrification and Crystallization 5

G2 q2 = 1 Ks–1 G1

15

Crystal (X)

0

Tg2

CP

10

–5

100

C px

40

L

→

C gp

60

→

–10 0

C lp

80

→

Cp /gtw K

glass, G Tg1

log τ

log η →

q1 = 105 K s–1

5

159

20 Tm

1/Tm

0

4

2 1/T →

100 200 300 400 500 600 700 800

Temperature K (b)

(a)

V cm3 mol–1

L 9.0 G1

Tg1 Tg2 ↓

↓

Tm

X

G2 8.5 500

1000

Temperature (c)

Figure 3.17. (a) Viscosity () as a function of reciprocal temperature showing the liquid to glass transitions at Tg1 and Tg2 for two different cooling rates, q = 1 and 105 K/s. The liquid to crystal, (X), transformation is also shown in the same figure. (b) Specific volume, V , and (c) specific heat, Cp , as a function of temperature showing step changes at the liquid to glass transition. CXp , Cgp and CLp refer to crystalline, glassy and liquid phases, respectively.

have been discussed in the following section with special reference to Ti- and Zr-based systems. 3.4.2 Thermodynamic considerations Pure metals are extremely difficult to vitrify under the conditions of rapid solidification which typically attain a cooling rate of about 106 K/s. Thin sections of

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splat quenched foils of Ni with dissolved gaseous impurities to a level of about 2% have been reported to vitrify. This, however, requires a cooling rate as high as 109 –1010 K/s, which is estimated to be attainable in thin sections ( 05 and x < 05, respectively. In the former case, a partitionless crystallization occurs as is reflected in faceted crystal/amorphous interfaces. Crystals of Zr 3 Fe Ni phase, which has a Re3 B type structure, tend to form a special orientation relationship with the core -crystals. Different crystallographic variants, which are twin related, are often found to form adjacent to each other, sharing the twin interface between them. The resultant morphology of the crystal aggregate, as illustrated in Figure 3.27(c), is described as the “sunflower” morphology. Electron diffraction patterns from individual “petals” and dark field imaging have revealed that the opposite petals have the same orientation and that each “petal” is twin related with the two adjacent petals. The growth of crystal aggregates with “sunflower” morphology is encountered quite frequently in Zr 3 Fex Ni1−x alloys with x > 05 in which polymorphic crystallization can occur. The growth of such three dimensionally symmetric aggregates can occur only if the matrix is fully

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isotropic as in the case of crystallization of an amorphous phase. From the consideration of the symmetry relation between the parent and the product phases, the amorphous to crystalline phase transformation is analogous to crystal formation from a liquid or vapour phase. This explains why the morphology of the crystal aggregates in the amorphous matrix often bears a similarity with the crystalline products from liquid or vapour phase. 3.4.5 Diffusion The thermal stability of metallic glasses is a subject of vital concern as one desires to produce glasses which will retain their amorphous nature at as high a temperature as possible. The current limit is about 1300 K, for some W-based glasses. Sometimes the amorphous phase is used as an intermediate which can be transformed into crystalline phases of desired grain structures. Such an approach is often adopted for the production of nanocrystalline structures using the amorphous phase as a precursor state. Phenomena such as diffusion, structural relaxation and crystallization need to be discussed for assessing the thermal stability of metallic glasses. A clear understanding of the changes occurring in metallic glasses during heat treatments is dependent on the elucidation of diffusion parameters and mechanisms. There have been very few direct measurements of diffusion in amorphous alloys. The experimental difficulties are considerable since the diffusion distance is very small for the accessible temperatures where crystallization can be avoided. When diffusion distances are in the range of 100 nm, techniques which permit composition analysis at a very high depth resolution need to be employed for measuring the concentration-depth profiles. These include Auger Electron Spectroscopy (AES) with sputter etching, Rutherford Backscattering Spectroscopy (RBS) and Secondary Ion Mass Spectroscopy (SIMS). Some indirect methods for the measurement of the diffusivities are also employed. In one of the early diffusion experiments, Gupta et al. (1975) determined the diffusivity of Ag in Pd81 Si19 . The surface of the alloy was sputter-deposited with 110Ag radioactive isotope. After diffusion annealing, the surface was sputter-etched. The Ag concentration was determined, from the radioactivity of the material removed, as a function of etching depth. One of the problems with such experiments is that different points on the surface are sputtered at different rates. Birac and Lesueur (1976) used a neutron beam, and the (n, ) reaction of 6 Li nuclei, for measuring its diffusion in Pd80 Si20 . Cahn et al. (1980) succeeded for the first time in measuring the self-diffusion of B in Fe40 Ni40 B20 . A layer of the same composition, containing 10 B and 11 B in the ratio, 96:4, was sputtered onto the metallic glass, which contained the natural isotopes in the ratio 20:80. After diffusion annealing, the surface was sputtered and 10 B/11 B ratio was measured by using SIMS.


177

Several indirect methods for the measurement of diffusivity are available. For example, it is possible to use the rate of primary crystallization. The diffusivities of B and C in Fe-B-C alloys have been deduced from measurements of the crystal growth rate (Koster and Herold 1980, Koster 1983). They obtained a diffusion coefficient, D, of 2 × 1019 m2 /s and an activation energy of 180 kJ/mol. These values led them to surmise that B diffuses as a substitutional atom rather than as an interstitial atom. This method has been applied to Fe40 Ni40 P14 B6 by Tiwari et al. (1981). There have been some attempts, by Taub and Spaepan (1979), to evaluate the diffusion coefficient from viscosity data. In the Pd–Si system, the diffusivity of gold in the as-quenched glass has been reported to be some orders of magnitude larger than the value of D as the glass is made to relax by annealing at a temperature below but close to Tg . These results strongly suggest that frozen-in ‘defects’ in metallic glasses, which are present in the as-quenched state, play an important role in the mechanism of diffusion. Let us focus our attention on Ti- and Zr-based metallic glasses which are primarily grouped under the class of metal–metal amorphous alloys with early and late transition metals as constituents. As has been indicated in Section 3.4.2, a fairly large number of binary alloys of this type are easy glass formers. Interpretations of diffusion data obtained in such systems are expected to be straightforward as complications due to multicomponent interactions will be absent. Measured values of diffusion coefficients, D, of different diffusing species in binary metal–metal amorphous alloys show an Arrhenius type dependence on temperature (Figure 3.28) when the data are replotted as log D against reciprocal of normalized temperature, Tg /T . This observation is indicative of the fact that for a given diffusing species and a given amorphous alloy, a single thermally activated diffusion mechanism remains operative over the entire temperature range studied. The influence of the atomic size of the diffusing species on the diffusion constant can be seen from the diffusivity data for Cu, Al, Au and Sb in a given metallic glass, Zr61 Ni39 . At any given temperature, it has been observed that DCu > DAl > DAu > DSb which is consistent with the fact that rCu < rAl < rAu < rSb where r is the respective atomic radius. The D values for Cu were found to be higher than the corresponding values for Al by about an order of magnitude in the temperature range of 556–621 K (Sharma and Mukhopadhyay 1990). The values of the activation energy, Q, for diffusion evaluated on the basis of the observed Arrhenius type temperature dependence of D, were found to be 1.33 ± 0.17 eV for Cu and 1.68 ± 0.13 eV for Al. The corresponding values of the preexponential factor, Do , were 10−757±146 and 10535±175 m2 /s, respectively. This comparison also reveals that the activation energy of diffusion in a given amorphous alloy increases with increasing atomic size of the diffusing species. Such a rule is expected to be


Diffusion Coefficient, D (m2 s–1)

178

10–18

10–22

10–26 1.0

1.2

1.4

Tg/T

Figure 3.28. Diffusion coefficients, D, of different diffusing species in binary metal-metal amorphous alloys as a function of normalized temperature Tg /T showing an Arrhenius type dependence on temperature.

valid only in cases where the diffusing species have more or less similar chemical interactions with the amorphous matrix. There have been a number of investigations to find out whether the diffusivities of a given species in an alloy in crystalline and amorphous states are significantly different. Contradictory results have been reported from these investigations. Valenta et al. (1981) have shown significantly slower diffusion of P and Fe in Fe40 Ni40 P14 B6 when the amorphous alloy is crystallized. Contrary to this, Akhtar et al. (1982a,b) have shown a much faster diffusion of Pt in Ni33 Zr67 after crystallization. Such contradictory results stem from the fact that there is a wide variety of crystallization mechanisms which result in a variety of crystal structures, phase distributions and grain sizes in the crystallized products. Diffusion data obtained from the homogeneous amorphous phase cannot be compared with that obtained in the crystallized product of the same material. Such a comparison is somewhat meaningful only in cases where the amorphous phase crystallizes into a single phase crystalline state with a large grain size (as in the case of polymorphic crystallization described in Section 3.5.1). Structural changes within the amorphous phase induced by heat treatments causing relaxation, plastic deformation and irradiation are expected to bring about changes in diffusivity in metallic glasses. Cantor and Cahn (1983) have reviewed the experimental data to arrive at the conclusion that diffusivity is very sensitive to relaxation in amorphous alloys which are very rapidly cooled through Tg .


179

Diffusion data reported in Zr61 Ni39 in the temperature range of 551–621 K show that a relaxation heat treatment does not affect diffusivity significantly (Sharma and Mukhopadhyay 1990). Autorelaxation of the glass during the cooling down from Tg appears to have reduced the influence of the subsequent relaxation heat treatment. Akhtar et al. (1982a,b) have reported the diffusion coefficient of Au in amorphous Zr67 Ni33 at four different temperatures using as-quenched, relaxed and plastically deformed specimens. At each temperature, the diffusivity in the deformed condition is found to be higher than that in the as-quenched condition, the latter in turn being higher than that in the relaxed condition. Cahn et al. (1982) have observed that Au diffusion coefficients in amorphous Zr64 Ni36 decrease after the amorphous alloy is irradiated with fast neutrons. They have argued that the irradiation-enhanced chemical short-range order causes a reduction in atomic volume with a corresponding reduction in the diffusion coefficient. Diffusion of small size H atom in Ti- and Zr-based metallic glasses assumes a great significance because of the possibility of H storage in some of these materials. The activation energy of H diffusion in Zr67 Pd33 has been reported to be 0.25 eV between 270 and 365 K, increasing to about 0.7 eV in the temperature range of 430–490 K. Similar characteristics have been observed in amorphous Ti–Cu alloys. Diffusion of H in hydrogenated Ti–Cu and Zr–Pd amorphous alloys is between one and two orders of magnitude faster than in the corresponding crystalline hydrides. Cantor and Cahn (1983) considered various experimental and theoretical information when available regarding diffusivities in amorphous alloys for arriving at possible atomistic mechanisms of diffusion in these systems. Since an Arrhenius type relation has been found to be valid for the temperature dependence of diffusivity in amorphous alloys, it is attractive to consider whether atomic models of diffusion in crystalline materials can also be applied to amorphous alloys. This approach can be further justified in view of the fact that the local arrangements of atoms in and the densities of amorphous and crystalline alloys are somewhat similar. The atom-vacancy exchange process is known to be the most important mechanism for both self- and impurity diffusion in crystalline alloys. In the absence of a reference lattice, a vacancy in an amorphous alloy can be defined as an empty space in the amorphous structure of atomic or near atomic dimensions. Several investigations have been made with a view to examining the stability of vacant sites of atomic dimensions in an amorphous alloy. From modelling work, it has been shown that if an atom is removed from the dense random packed structure of an amorphous alloy, atomic vibrations quickly redistribute the excess space

180


over a large volume. This tendency of smearing the excess volume created by the removal of an atom does not allow the presence of a near atomic size vacant space in the amorphous structure. The structure and size of soft sphere dense random packed models of amorphous alloy structures show that most interstitial sites are surrounded by distributed tetrahedral and octahedral groups of atoms with more tetrahedral and fewer octahedral sites than in an equivalent close-packed crystal. The size distribution in Figure 3.29 shows that a small fraction of octahedral interstices have sizes in the range of 0.6–0.7 of the atomic diameter, and these large interstices can be considered as vacancies in the amorphous structure. 3.4.6 Structural relaxation The structure and properties of a glass depend on the quenching rate. When a glass is annealed, its structure will first relax to that of a glass formed at lower cooling rates and ultimately tend towards that of an “ideal” glass. Egami (1983) used energy dispersive XRD methods and showed that two types of change occurred during relaxation. One is related to the topological short-range order. During the process of atomic movement, the tetrahedra per se are not affected by their relative configurational change. A higher degree of topological short-range order is established by a highly collective phenomenon involving the cooperation of a number of atoms. The second change is related to the chemical short-range ordering during which site interchange of atoms takes place.

Fraction of sites

0.3

Tetrahedral

0.2

octahedral – fcc interstitial radii

Radius of vacant sites in amorphous structure

0.1

0.2

0.4

0.6

Vacant site radius/atomic radius

Figure 3.29. Size distribution of vacant sites in amorphous structure.


181

Further studies have indicated that relaxation mechanisms can be divided into reversible and irreversible types. Reversible changes appear to be associated with changes in the chemical short-range order. Kurusmovic and Scott (1980) have shown that the Young’s modulus of the Fe40 Ni40 B20 glass may be cycled reversibly between the values which are characteristic of different annealing temperatures. The analogy with short-range order in crystalline alloys is particularly striking. The irreversible change appears to be associated with a change in the topological short-range order. During structural relaxation, the alloy becomes denser. Significant changes occur in many properties such as the electrical resistivity, magnetic anisotropy, Curie temperature, elastic modulus and mechanical properties. Some changes are beneficial, while others are detrimental. The changes in magnetic and mechanical properties are discussed below. Luborsky (1983) have reported remarkable improvements in the magnetic properties of Fe–Ni-based glasses after stress-relief annealing for 2 h at 100 K below their glass transition temperature. These changes include an increase in remanence, a decrease in the saturation field and a change in the Curie temperature. However, if the treatment leads to crystallization, then the magnetic properties deteriorate. Hence, it is important to ensure that the compositions of metallic glass ferromagnets are chosen so as to have good thermal stability. The Curie temperature can be determined either by magnetic measurements or by locating the appropriate thermal anomaly in a differential scanning calorimeter (DSC) run. Egami (1983) found that cycling Fe27 Ni53 P14 B6 repeatedly between 523 and 573 K caused the Curie temperature to cycle between 368 and 375 K. These reversible changes are due to a change in the position of Fe and Ni atoms and the consequent alteration in the chemical short-range order. Further exploration of the use of low-temperature annealing in bringing about such beneficial changes in magnetic properties appears to be desirable. One of the attractive properties of metallic glasses is their large ductility in bending and compression. In many cases, low-temperature annealing treatments lead to the loss of this ductility. There have been several investigations of this temper embrittlement phenomenon, and it appears to be dependent on the composition. In the case of Fe40 Ni40 P14 B6 , a heat treatment at 373 K for 2 h leads to embrittlement. AES has been used for demonstrating that segregation of P occurs during structural relaxation which leads to embrittlement (Walter and Bertram 1978, Walter 1981). In general, metal–metal glasses and Cu60 Zr40 appear to be immune to this type of embrittlement. An interesting suggestion with regard to the embrittlement tendency has come from the group of Davies (1978). Alloys which eventually crystallize as an fcc phase do not exhibit the embrittling tendency, whereas alloys which crystallize

182


into a bcc or hcp phase become brittle on relaxation. This implies that the structural groupings of the crystalline phases responsible for embrittlement can be partially observed in the amorphous phase. This might also explain why embrittlement occurs within a certain composition range in a given system. 3.4.7 Glass transition The nature of the glass transition is still a matter of controversy. Experimental evidence and theoretical models suggest the glass transition to be a first-order phase transition, based on the free volume approach, a higher order phase transition or no phase transition at all, e.g. kinetic freezing. However, there is a general agreement that the maximum undercooling of a liquid is limited to the isentropic temperature in order to avoid the paradoxical situation described by Kauzman (1948) where the configurational entropy of the supercooled liquid becomes smaller than the configurational entropy of the ordered equilibrium phase. Consequently, as long as crystallization can be prevented, the undercooled liquid will freeze to a glass close to the ideal glass transition temperature, Tgo , where the entropy difference between the liquid and the equilibrium crystalline phase would vanish. In reality, the glass transition sets in at a temperature somewhat above Tgo . Depending on the deviation of the glass transition temperature, Tg , from the ideal glass transition temperature, Tg , the glass attains different relaxation states. Only at an infinitely slow cooling rate, if the liquid is vitrified (by avoiding crystallization), the liquid to glass transition occurs at Tgo and the resulting glass attains a fully relaxed state. Such a transformation can be considered as a second-order transition at Tgo . Under realistic cooling rates, a glass having excess entropy and consequently not in the fully relaxed state forms at Tg . Depending on the extent of relaxation of the product glass, the glass transition temperature measured from experiments varies with the imposed cooling rate, a higher cooling rate yielding a higher value of Tg , as shown in Figure 3.17 in which Tg1 and Tg2 are the glass transition temperatures for the cooling rates, G1 and G2 (G1 > G2 ), respectively. The glass transition event is also encountered during heating experiments. Continuous heating experiments in a DSC often show an endothermic event prior to the large exothermic event of crystallization. A thermogram obtained during heating of the Zr-35 at.% Ni glass at a heating rate of 5 K/min clearly reveals the endothermic event attributable to the glass to liquid transition preceding the crystallization event (Figure 3.30). Taking into account experimental values of the specific heat of a stable and highly undercooled liquid and measured values of the enthalpy of crystallization of the amorphous alloy, the undercooled liquid at Tg is found to exhibit only a very small excess entropy in comparison with the stable crystalline phase. Specific heat, thermal expansion and Mossbauer spectroscopy data on several fully


183

Heat flow exothermal 50 000 mW

↓

150

250

350

400

Temperature, °C

Figure 3.30. DSC thermogram of Zr-35 at.% Ni glass obtained at a heating rate of 5 K/min showing the endothermic nature of the glass to liquid transition.

relaxed amorphous alloys reveal that the glass transition can be approached under internal equilibrium conditions (Tg approaching Tgo and becoming independent of the heating rate). Usually amorphous alloys form in composition ranges where the heat of mixing between the components is negative (strong ordering tendency). However, there are instances where a positive deviation from the ideal solution behaviour is noticed in narrow composition ranges where the amorphous phase tends to separate into two phases, both having amorphous structure. Such a system is expected to exhibit two glass transition temperatures. Experimental observations of two glass transition events and of a phase separated microstructure of the amorphous alloy have led Tanner and Ray (1980) to infer the presence of a two-phase amorphous structure in the Zr36 Ti24 Be40 alloy. Using an atom probe microscope, Grunse et al. (1985) have shown the presence of spatially extended concentration waves in as-quenched Ti50 Be40 Zr10 . Observations on structural relaxation and localized fluctuations in structure and composition of metallic glasses have prompted a model of glass transition based on the heterogeneous glass in terms of density and concentration. In this model, it is visualized that a glass consists of liquid-like regions of large free volume or high local free energy and solid-like regions with small free volume or low

184

Phase Transformations: Titanium and Zirconium Alloys τ*

(a)

Relaxation time (τ)

↓

Relaxation time (τ)

↓

T1 > Tg

Frequency

Frequency

T2 < Tg

(b)

Figure 3.31. Schematic diagram showing relaxation spectra at (a) temperatures above Tg and (b) temperatures below Tg .

free energy. Each region undergoes a transition at a frequency much smaller than the Debye frequency (∼1013 s−1 ) between local energy minima corresponding to different configurational states, the relaxation time, i, being proportional to exp− i /kB T ) where i is the energy barrier between these states. The relaxation spectra at temperatures below and above Tg are schematically shown in Figure 3.31. At T1 > Tg , the whole spectrum lies to the left of the time of measurement ∗ (for example, 30 s), so that the whole system undergoes frequent configurational transformations and is liquid-like. With lowering of temperature below Tg , (T2 < Tg ), the whole spectrum shifts to longer times such that relaxation times for most of the regions are greater than ∗ , and the isolated liquid-like regions are small in volume fraction and are embedded in the rigid solid matrix. At such temperatures, localized, short-range structural relaxation can occur, leading to glass transition in these small domains. In the close vicinity of Tg , the peak of the distribution of the relaxation time is located near ∗ , leading to a long-range, cooperative structural relaxation which causes a rapid decrease in the relaxation time and an accompanying rise in the viscosity.

3.5

CRYSTALLIZATION

Several experimental techniques have been used to monitor the crystallization of metallic glasses. Among these, DSC and TEM have proved to be particularly useful. In the case of DSC, crystallization gives rise to distinct exothermic peaks. The heat of crystallization can be measured and is found to be of the order of 40% of the heat of fusion of the alloy, the remaining enthalpy having been extracted from the liquid during quenching. Using electron microscopy, the morphology


185

and structure of crystals can be established and the mechanism and kinetics of crystallization can be followed. Heating of an amorphous alloy leads to several other changes apart from relaxation. These include glass–liquid transition, phase separation and crystallization. Not many studies have been made on the glass–liquid transition. Although phase separation into two amorphous phases is a well-documented event in the case of oxide glasses, metallic glasses appear to undergo phase separation in only rare instances. Evidence for phase separation has been reported in Pd74 Au8 Si18 by Chou and Turnbull (1975) and in B40 Ti24 Zr36 by Tanner and Ray (1979). DSC plots indicate the occurrence of two glass transition temperatures. Banerjee (1979) reported a spinodal decomposed amorphous microstructure in the Zr-24% Fe alloy. Piller and Haasen (1982) have used the sensitive atom probe field ion microscope in order to demonstrate that Fe40 Ni40 B20 decomposes into two amorphous regions, one having composition corresponding to (Fe,Ni)3 B and the other being a B-deficient region. 3.5.1 Modes of crystallization From symmetry rules, it can be shown that the amorphous to crystalline phase transition is necessarily a first-order transition. This is consistent with the observed nucleation and growth processes encountered by different investigators studying crystallization. As there is no periodic arrangements of atoms in the parent amorphous structure, there is no possibility of achieving a lattice correspondence between the parent and the product structures. Therefore, the occurrence of crystallization via “military” atom movements is ruled out. Hence, the transition is expected to occur essentially by diffusional atom movements. Depending on the diffusion distances involved in the crystallization process, one can broadly classify the mechanisms of crystallization into three broad categories, namely (a) polymorphic crystallization, (b) eutectic crystallization and (c) crystallization involving long-range diffusion – primary followed by eutectic or primary followed by polymorphic crystallization. These three modes of crystallization are schematically illustrated in Figure 3.32. (a) Polymorphic crystallization (A → ): When the compositions of the parent amorphous phase and of the product crystalline phase are the same, the crystallization process involves diffusional atomic jumps across the advancing transformation front. This situation is analogous to that in massive transformation or in “partitionless solidification” processes and is illustrated in Figure 3.32 for the alloy composition c1 . Koster and Herold (1980) have termed this type of crystallization as “polymorphic crystallisation”.


Free energy (G)

186

Gβ

GA

Gα

c1 c4 c6

A

c2 c3

c5

c7

B

c, atom fraction of B A

A α

α

A′

c→

c5 c1

A

α

c2 A

Distance → A (c1)→ α (c1)

Polymorphic (c 1)

E

α E

A

α

E

α

c4 Distance → A → α (c4) + A′ (c5)

Distance → A′ (c5) → α (c6) + β (c 7)

Primary + Eutectic (c 2)

c3

E A

A

Distance → A (c3) → α (c6) + β (c7) Eutectic (c 3)

Figure 3.32. Schematic representation of different modes of crystallization: polymorphic, primary followed by eutectic and eutectic occurring in amorphous alloys of compositions given by c1 , c2 and c3 , respectively. Free energy changes which motivate the crystallization process are indicated by arrows in the free energy – concentration (G–c) plots corresponding to the amorphous (A) and the two crystalline phases, and . In polymorphic crystallization, the amorphous alloy of composition c1 transforms into the -phase of the same composition. In primary crystallization, phase of composition c4 forms first from the amorphous alloy of composition c2 ; the composition of the latter gradually changes to c5 which finally decomposes into an eutectic mixture (E) of and . In eutectic crystallization, the amorphous phase of composition c3 directly transforms into the eutectic mixture (E) of and . Concentration profiles across a crystalline particle is shown below each of the schematic micrographs.

(b) Eutectic crystallization (A → + ): The partitioning of alloying elements into two crystalline phases (illustrated in Figure 3.32 for the composition c3 ), which are forming simultaneously from the parent amorphous phase in a cellular transformation, requires diffusion at or near to the transformation front. Eutectic/eutectoid decomposition and cellular precipitation are the analogous phase transformations in crystalline systems.


187

Eutectic crystallization, which has been encountered in several systems such as Fe–B, Fe–Ni–B, Mo–Ni, and Zr–Fe, occurs at a relatively low rate as compared to polymorphic crystallization. Decomposition of an amorphous phase into a mixture of a crystalline and a second amorphous phase, in a cellular transformation mode, appears to be possible in a system in which the amorphous phase has a tendency towards phase separation (or unmixing). However, in the case of metallic glasses, such a transformation, which is analogous to the monotectoid reaction, has not been encountered. (c) Primary crystallization followed by eutectic or polymorphic crystallisation: During the formation of a crystal having a composition which is different from that of the parent amorphous phase, a long-range diffusion field is established ahead of the transformation front. Primary crystallization, as designated by Koster and Herold (1980), involves such a process, the kinetics of which are generally controlled by the mechanism of long-range atom transport in the amorphous matrix. The composition of the matrix amorphous phase may finally transform via one of the many possible phase reactions. In the case of Fe–B, the amorphous matrix gradually attains the Fe75 B25 composition and then transforms into the Fe3 B phase via a polymorphic crystallization process. By analogy with various liquid-to-crystal phase reactions, one can visualize many possible reactions, such as the following: (a) Primary crystallization followed by a peritectic/peritectoid reaction (A→ (primary) + A → ): There exists a possibility of this transformation sequence which has not been reported in any metallic glass system. (b) Primary crystallization followed by an eutectic reaction (A → (primary)) + A → (primary) + + (eutectic): A and A are amorphous phases having different compositions, and and are different crystalline phases as shown in Figure 3.32 for composition c2 . The appearance of more than one exothermic peak in DSC thermograms appears to originate from such successive phase reactions (Figure 3.33).

3.5.2 Crystallization in metal–metal glasses Metal–metal glasses have not received the same degree of attention as metal– metalloid glasses, although this situation appears to be changing with increasing interest in these glasses. Metal–metal glasses present several features of interest. Unlike metal–metalloid glasses, which are generally restricted to compositions of around 20 at.%, metal–metal glasses can be formed over a wider range of compositions. Both metal–metalloid and metal–metal glasses can be prepared around compositions corresponding to deep eutectics. In addition, metal–metal

188

Phase Transformations: Titanium and Zirconium Alloys Eutectic or polymorplic

·

Rate of heat flow (H )

Primary Crystallization Glass transtion

2 Tg

Tx Temperature

Polymorplic or eutectic crystallization

Tg

1 T x1

T x2

Temperature Primary followed by eutectic or polymorplic crystallization

Figure 3.33. Differential scanning calorimetry thermograms showing, dH/dt, the rate of heat flow versus temperature, T , at a constant rate of heating (dT /dt = constant) for (a) polymorphic and eutectic crystallization and (b) primary followed by eutectic or polymorphic crystallization. The endothermic event of glass transition at Tg and the exothermic events of crystallization at Tx are indicated. While for polymorphic and eutectic crystallization a single exothermic event is observed, as shown in (a), two distinct thermal events are noticed in primary crystallization followed by either eutectic or polymorphic crystallization as shown in (b).

glasses can form at compositions which correspond to stoichiometric compounds having high melting points. There are also indications of structural differences. Polyhedral packing, characterized as Kasper polyhedra, appears to be a dominant structural feature in metal–metal glasses. Systems which have been investigated in detail include Cu–Zr, Ni–Zr, Fe–Zr, Ni–Nb, Ti–Be–Zr and Mg–Zn. In order to illustrate the type of investigations made on these alloys, the Ni–Zr and Fe–Zr systems are used as examples in the discussion which follows. The equilibrium diagram for the Ni–Zr system contains four well-defined eutectics at 8.8, 36.3, 63.5 and 75.9 at.% Zr. Glasses are formed at compositions which correspond to these eutectics as well as at compositions which correspond to the equilibrium intermetallic phases. The results of Dong et al. (1981) and Dey et al. (1986) with regard to four alloys are described below. The Ni365 Zr635 alloy corresponds to an eutectic between the Zr2 Ni and ZrNi phases. Crystallization of this amorphous alloy involves two steps: primary crystallization of Zr2 Ni crystals, followed by the formation of ZrNi crystals. The activation energy for the nucleation of primary crystals has been reported to be 500 kJ/mol, and the diffusion controlled growth of these crystals of average diameter, d, has been shown to be characterized by the relationship, d t1/2 where t is the annealing time.


189

The Ni333 Zr667 alloy exhibits polymorphic crystallization, which results in the formation of Zr2 Ni crystals. The activation energy associated with the crystallization process has been determined to be about 320 kJ/mol from both DSC experiments performed in the continuous heating mode and measurement of crystal size as functions of time and temperature. Dey et al. (1986) have observed the presence of very closely spaced planar faults within the Zr2 Ni crystals (Figure 3.34). The Zr72 Ni28 alloy crystallizes into an off-stoichiometric Zr2 Ni phase which has a C16 (tetragonal) structure. The crystals exhibit a dendritic morphology but a closer examination reveals a spherulitic morphology. The activation energy for growth has been measured to be 180 kJ/mol. DSC experiments carried out on Zr76 Ni24 have revealed two exothermic peaks. The first has been attributed to the formation of the hcp -phase, identified by TEM studies. Using peak shift observations obtained from DSC runs made at different heating rates, the activation energy associated with the primary crystallization event has been determined to be 310 kJ/mol. This value agrees closely with that observed in Ti50 Be40 Zr10 , where primary crystallization into has been reported. As the -phase is expected to be solute lean, the rate-controlling process is identified as being the diffusion of the solute element in the amorphous matrix. The second step in the crystallization process could be either an eutectic reaction, leading to simultaneous formation of and Zr2 Ni, or a polymorphic reaction, leading to the formation of Zr2 Ni. The activation energy associated with the second step has been found to be 180 kJ/mol (Table 3.5). Such a low value of the activation energy is consistent with the occurrence of a eutectic or a

Figure 3.34. Bright field TEM micrograph showing the presence of very closely spaced planar faults within Zr2 Ni crystal.

190

Composition

Sequence (mode)

Bulk or surface

Zr76 Fe24

(1)

Bulk

Zr76 Fe24 Ni4

(2)

Bulk

Zr76 Fe16 Ni7

Tg 80 K/min

T1 (K) 20 K/min

Tp (K) 20 K/min

E/En /Eg (kJ/min)

T (K)

n/n1 /n2

650.0

653.5

656.0

272.0 (E)/545.0 (En )/168.0 (Eg )

626.0

3.10 (n)

641.0

653.0

655.0

286.0 (E)

631.0

2.71 (n)

639.0

650.7

652.0

278.0 (E)

634.0

2.65 (n)

Zr76 Fe12 Ni12

(3)

Bulk

627.0

646.0

648.0

275.0 (E)

633.0

2.55 (n)

Zr76 Fe8 Ni16

(4)

Surface

624.0

634.0

647.0

274.0 (E)

631.0

Zr76 Fe4 Ni20

(5)

Bulk

624.0

634.0

647.0

236.0 (E)

634.0

Zr76 Ni24

(7)

Bulk

650.0

652.0

654.0

271.0 (E) 410.0 (En )

631.0

2.21 (n1 )/ 2.73 (n2 ) 1.98 (n1 )/ 4.00 (n2 ) 3.20 (n)

(1) A → Zr3 Fe (polymorphic); (2) A → Zr3 (Fe,Ni) + A (primary) → + Zr2 Ni (eutectic); (3) A → Zr3 (Fe,Ni) (polymorphic); (4) A → Fe rich (Fe,Ni)3 , Zr(Ll2 ) + A (primary); (5) A → Zr3 (Fe,Ni); (6) A (primary), A → Zr2 Ni + (eutectic); (7) A → Zr2 Ni + (eutectic).


Table 3.5. Crystallization sequence (mode), glass transition temperature, Tg , crystallization temperature, Tx , peak crystallization temperature, Tp , activation energies for (i) overall crystallization, E; (ii) nucleation, En , and (iii) growth, Eg , isothermal annealing temperature, T and Avarami exponent for (i) single-step process (n) and (ii) two-step processes (n1 and n2 ).


191

polymorphic crystallization process, both of which involve only short-range atom transport at the transformation front for the distribution of the solute into the two product phases. However, Dong et al. (1981) have reported single step polymorphic crystallization. Such differences can arise due to variations in the initial conditions of the amorphous phase (e.g. the extent of relaxation which the glass has undergone during the melt-spinning operation). Koster and Herold (1980) have reported that polymorphic crystallisation occurs in Fe40 Zr60 , Fe30 Zr70 and Fe24 Zr76 glasses. The activation energy associated with the growth of crystals has been found to be about 170 kJ/mol. The resulting crystals in the aforementioned alloys are FeZr2 , FeZr2 and FeZr3 respectively. Dey and Banerjee (1986) have carried out DSC experiments employing both isothermal holding and continuous heating runs and have found that the activation energy associated with the overall crystallization process in the Fe24 Zr76 glass is 270 kJ/mol. TEM studies have revealed the presence of Zr3 Fe crystals (orthorhombic Re3 B structure), with planar boundaries separating the crystalline and amorphous phases. This is possible because of the absence of any long-range atom transport during the polymorphic crystallization process. Crystal aggregates of a fascinating shape have been observed in both partially crystalline as-melt-spun tapes and crystallized samples of fully amorphous Zr76 Fe24 glass. These crystal aggregates, which presumably had formed at a very early stage in the crystallization process, consisted of six petals originating from a central spherical crystal, thus giving rise to a “sunflower”-like appearance (Figure 3.35).

Figure 3.35. Bright field TEM micrograph showing crystal aggregate having a “sunflower” morphology.

192


Crystallographic analyses of each of these petals and of the core have permitted the orientation relationship to be determined. The petals consisted of an ordered phase crystals of Zr3 Fe which can be viewed as being an ordered structure based on

-Zr, while the core had a bcc structure. The formation of such an agglomerate is energetically favourable because the unique orientation relationship of the adjacent crystals permits the formation of low energy interfaces between them. 3.5.3 Kinetics of crystallization It has been mentioned earlier that the amorphous to crystalline transition (devitrification) occurs in different modes such as polymorphic, eutectic and primary followed by eutectic. The kinetics of the process is, therefore, governed by the mode which operates in a given system. Devitrification, being a strongly first-order transition, occurs by the nucleation of crystals in the amorphous matrix followed by the growth of these nuclei by the movement of the crystal/amorphous interface, resulting in a progressive consumption of the matrix amorphous phase. The overall kinetics of devitrification is determined by the number density of quenched-in nuclei, the rate of thermally activated fresh nucleation and the rate of growth of the crystalline phase. If the nucleation rate is so high at the early stages of the transformation that all the quenched-in nuclei are consumed before appreciable growth occurs and thermally activated fresh nucleation is limited, the number density of crystalline particles will remain more or less constant and their sizes will remain uniform during the growth process. On the other hand, when fresh nucleation continues along with growth, a distribution of particle sizes will result. As mentioned earlier, the composition of a growing crystal or the average composition of a two-phase nodule, which are the products of polymorphic and eutectic crystallization respectively, remains the same as that of the amorphous matrix during the growth process. Therefore, in these cases, there is no longrange concentration field ahead of the growing crystals. In contrast, if the growing crystalline phase has a composition different from that of the amorphous matrix (as in the case of primary crystallization), a long-range diffusion field is created ahead of the crystal/amorphous interface. It is possible to identify the mode of crystallization from the analysis of the kinetics of the overall process (consisting of nucleation and growth) using the Johnson, Mehl and Avrami (J–M–A) formulation (Burke 1965) which relates the actual fraction, f , of the transformed volume with t, the duration of the transformation: f = 1 − exp−Ktn

(3.39)


193

where n, known as the Avrami exponent, assumes different values for different mechanisms and geometries of the growing crystals, as discussed later in this section. The temperature dependence of K is given by the Arrhenius equation: Q (3.40) K = Ko exp − RT where Q is the activation energy of the process. Data on the fraction transformed at a given temperature for different durations of the transformation can be analysed to obtain the value of n for an isothermal crystallization process in a glass. It may be noted that it is not possible to determine unequivocally the nucleation and growth behaviour of a crystallization process from data on the time dependence of the transformed volume only, as is often attempted. It is essential to have additional knowledge about the process, best gained from direct microscopic observations on the growing crystals as a function of time. Table 3.6 shows the values of the Avrami exponents, n, for different types of nucleation and growth transformations. DSC is often used for studying the crystallization kinetics mainly because of the precise control of temperature and heating rates and the high sensitivity of recording events of heat release associated with this technique. Both continuous heating and isothermal holding experiments are carried out for gaining information regarding the crystallization kinetics. The methodology of this analysis has been discussed later in this section for illustrating typical cases representing polymorphic, eutectic and primary crystallization. In the context of devitrification of metallic glasses, it is to be noted that during continuous heating experiments, crystallization occurs at temperatures only slightly higher than the glass transition temperature, Tg . Since structural relaxation of the Table 3.6. Values of Avrami exponent, n, for different types of nucleation and growth transformation. Geometry

Nucleation rate

n

Interface controlled Plate Cylinder Sphere Sphere

Rapid, depleting Rapid, depleting Rapid, depleting Constant

1 2 3 4

Long-range diffusion controlled Sphere Sphere Cylinder Plate

Rapid, depleting Constant Rapid, depleting Rapid, depleting

3/2 5/2 1 1/2

194


glass occurs during heating it up to Tg , causing substantial reductions in atomic transport rates, the nucleation and growth rates of crystals depend not only on the temperature but also on the thermal history. In continuous heating calorimetric studies, when the temperature is increased linearly with time, the thermal effects of glass transition and devitrification may overlap, making the analysis of the results very difficult. It is important, therefore, to select such systems for studies on devitrification kinetics for which the kinetic crystallization temperature is several degrees celsius above Tg . In isothermal kinetics studies, the occurrence of a suitably long incubation period prior to detectable transformation makes it convenient to record the thermal evolution data on a stable baseline. Isothermal DSC results can be analysed to obtain the fraction transformed, ft, as a function of time. The zero time is defined by the instant when the isothermal holding temperature is reached. The total heat evolution due to crystallization is reflected in the exothermic peak observed during the isothermal holding at a given temperature. The measurements of the area under the heat evolution curve up to different time periods give the ft values which, when presented in a (J–M–A) plot of ln− ln1 − f ) versus ln t, gives a straight line fit, the slope of the straight line giving the J–M–A exponent, n. This method of analysis of results of isothermal kinetics experiments in DSC has been illustrated later with examples of polymorphic crystallization in binary Zr76 Fe24 and primary crystallization in Zr67 Ni33 glasses. Even though the interpretation of isothermal kinetics data is straightforward in principle, a number of problems are encountered in practice. It is often difficult to maintain the baseline flat over the entire length of the transformation time. The variation in the specific heat (Cp ) of the amorphous and crystalline phases contributes to the baseline shift, which can be computed by taking into account the Cp value of the mixture of the amorphous and the crystalline phases prevailing at different stages of the transformation. The temperature range over which isothermal experiments can be conducted is limited: if the temperature is too high, the transformation may start even before the isothermal holding is reached and will certainly overlap instrumental transients; if it is too low, the rate of reaction is so sluggish that the fraction transformed cannot be determined accurately. For these reasons, and also because of its greater speed and convenience, non-isothermal DSC (such as continuous heating experiments) is often practiced while conducting studies on kinetics of crystallization. This technique is also useful in cases where the crystallization process exhibits more than one exothermic peak, suggesting the occurrence of more than one crystallization event, such as primary followed by eutectic crystallization. In a continuous heating experiment, the differential power required for maintaining the temperature of a sample and a reference material is measured as a function of temperature, the heating rate being kept constant.


195

Essentially such an experiment yields the rate of enthalpy change of the sample undergoing a first-order phase transition as a function of temperature. An outline of the reaction kinetics of a phase transformation under a constant heating rate condition, as worked out by Kissinger (1957), is presented here. The kinetics of a solid state reaction can be described by the equation df = A1 − f n exp−Q/RT dt

(3.41)

where df /dt is the rate of change in the fraction transformed, n is the order of the reaction and Q is the activation energy. As the temperature is raised at a constant rate, = dT/dt, the transformation rate, df /dt, rises to a maximum value at T = Tp and then falls due to a continuous reduction in the untransformed volume, (1 − f ), which becomes zero at the completion of the process. By differentiation of Eq. 3.41 d df Q df n−1 = − An1 − f exp−Q/RT 2 dt dt RT dt (3.42) d df AtT = Tp =0 dt dt Therefore, the maximum transformation rate and Tp , the temperature at which this rate reaches the peak value, are related by Q = An1 − f n−1 exp−Q/RTp RTp2 This can be reduced to

Tp2

(3.43)

= A exp−Q/RTp

(3.44)

A is a temperature-independent factor, provided the fraction transformed corresponding to the peak transformation rate remains the same at all temperatures. Henderson (1979) has shown that the peak in df /dt occurs at f = 063 for J–M–A kinetics and linear heating. Equation (3.44) suggests that a plot of ln/Tp2 versus 1/Tp (Kissinger plot) can be expected to yield a linear fit, the activation energy of the overall transformation process being given by the negative slope of the plot. Experimental determination of the activation energy of the overall crystallization process can, therefore, be made by recording DSC thermograms at different heating rates ( ), which give Tp values for different . The usefulness of the Kissinger method for studying kinetics of crystallization is illustrated in Section 3.6.1 by considering examples of polymorphic, eutectic and primary + eutectic crystallization modes in Zr–Fe–Ni glasses.

196


The activation energy, Q, determined from the Kissinger peak shift method under the continuous heating condition refers to the activation energy of the overall process which includes both nucleation and growth. Ranganathan and Heimendahl (1981) have proposed a methodology for separately determining the activation energies associated with the nucleation and growth processes, using experimental data from isothermal kinetics studies. For considering the kinetics of the nucleation and growth processes separately, let us take the case of a constant nucleation rate, I, which can be expressed as

Q I = Io exp − n RT

(3.45)

where Qn is the activation energy for nucleation, which is a sum of W ∗ , the energy required to form a critical nucleus and Qd , the activation energy of diffusion. Qn can be determined from the slope of a plot of ln I against 1/T . TEM examinations of samples aged for different durations at a few selected temperatures can provide data on the nucleation density as a function of time. The activation energy for nucleation of crystals can be determined using such data analysed on the basis of Eq. 3.45. The measurements of the size of crystals (for polymorphic and primary crystallization) or of nodules (for eutectic crystallization) can provide data which can be used for obtaining the activation energy for growth. The radius, r (or a representative linear dimension), of crystals or nodules growing with time, t, following a linear growth law r = A1 t

(3.46)

is pertinent to polymorphic and eutectic crystallization, while the growth is parabolic r = A2 Dt1/2

(3.47)

for primary crystallization which is usually bulk diffusion controlled. The growth rate, u= dr/dt, can be written as

Qg u = uo exp − RT

(3.48)

where Qg is the activation energy for growth, which can be determined from the slope of the plot of ln u against 1/T .


197

An illustrative example of such a kinetic analysis of nucleation and growth processes separately from the experimental data obtained from isothermally treated samples is shown in Figure 3.36. Isothermal holding of metallic glass samples in a DSC close to the crystallization temperature, and monitoring the rate of heat evolution as a function of time, gives data on the fraction transformed as a function of time which can be analysed using the J–M–Avrami formulation (Eq. 3.39) for obtaining the Avrami exponent. The method of this analysis for polymorphic crystallization in Zr-33 at.% Ni glass is shown in Figure 3.37. Let us now consider the relationship between the activation energies for the overall process (Q) and for the nucleation (Qn ) and the growth (Qg ) steps for the following situations: Case (i): Nucleation rate, I = 0 and linear growth, u = constant. This situation arises when a fixed number (N ) of quenched-in nuclei operate and no fresh thermally activated nucleation occurs. A linear growth rate is a characteristic feature of polymorphic and eutectic crystallization in which atom movements are essentially confined to the vicinity of the transformation front. If the growing particles (for polymorphic) and nodules (for eutectic) are assumed to be spherical, the fraction transformed, f , can be expressed as 4 3 3 (3.49) f = 1 − exp − Nu t 3 From Eqs. 3.39, 3.40, 3.48 and 3.49, we get the Avrami exponent, n = 3 and Q = 3Qg . In the case of two dimensionally growing particles, i.e. discs with fixed thickness, the same analysis could be applied with n = 2 and Q = 2Qg . For the case of one-dimensional growth (needles) n = 1 and Q = Qg . Case (ii): Constant nucleation rate, I > 0 and linear growth, u = constant. In this case, the fraction transformed can be written for spherical particles as (3.50) f = 1 − exp − u3 It4 3 Substituting the values of I and u from Eqs. 3.45 and 3.48, we obtain Qn + 3Qg 4 3 t f = 1 − exp − Io uo exp − 3 RT Therefore Q = Qn + 3Qg and the Avrami exponent, n = 4.

(3.51)

198


5.0

623 K 40

4.0

Max diameter (μm)

623 K

623 K Number of nuclei (μm)–3

Zr76Fe24 Zr76Ni24

Zr76Fe24 Zr76Ni24

30

623 K 20

623 K

10

0

100

200

3.0

593 K 2.0

1.0

623 K

0

580 K

603 K 623 K 613 K 590 K 600 K 610 K

0.0

300

0

100

200

Time (min)

Time (min)

(a)

(b) 1.0

300

Zr76Fe24 Zr76Ni24

0.0

–1.0

Max crystal size (μm)

0.8

605 K 595 K

0.6

In ISS In G

615 K

–2.0

–3.0

ISS

0.4

G ISS

–4.0

0.2

G 0

4

8

12

16

Time1/2 (min1/2)

(c)

20

24

–5.0 135

150

160

170

1/T (10–3 K–1)

(d)

Figure 3.36. Kinetic analysis of nucleation and growth processes from the experimental data obtained from isothermally treated samples at different temperatures: (a) number of nuclei as a function of time, (b) maximum diameter as a function of time, (c) crystal size as a function of time in case of primary crystallization and (d) plots of nucleation rates and growth rates against 1/T for Zr76 Fe24 and Zr76 Ni24 . Activation energies of the nucleation and growth processes are determined from these plots.


199

0.0 679 K

1.0

677 K

675 K 673 K

–0.1 672 K

677 K 675 K

679 K

673 K 672 K

–0.2 –0.3

0.9

–0.4

0.8

Log (– log(1 – x ))

–0.5 0.7

x

0.6 0.5 0.4

–0.6 –0.7 –0.8 –0.9 –1.0

0.3

–1.1 –1.2

0.2

–1.3 0.1 0

–1.4 1

2

3

4

5

6

7

8

9

–1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time (min) (a)

Log t (b)

Figure 3.37. (a) Fraction transferred as a function of time for polymorphic crystallization in Zr-33 at.% Ni. (b) Plots drawn to estimate the Avrami exponent using the fraction transformed data.

This case also applies to the primary recrystallization process of cold worked metals and alloys. Case (iii): Nucleation rate, I = 0 and parabolic growth, r = A2 Dt1/2 When N number of spherical quenched-in nuclei are operating, the fraction transformed at time, t, is given by 4 3 3/2 15Qd f = 1 − exp − NA2 Do exp − (3.52) 3 RT Following a similar procedure as the one for cases (i) and (ii), we get Q = 3/2Qd and n = 3/2. For a two dimensionally growing particle, n = 2/2 = 1 and Q = Qd Case (iv): Constant nucleation rate I > 0 and parabolic growth, r = A2 Dt1/2 For three-dimensional growth of spherical particles in duration t, 4 8 t f = ∫t=0 I A32 D3/2 t3/2 dt = IA32 D3/2 t5/2 3 15

(3.53)

200


3.5.4 Crystallization kinetics in Zr76 Fe1−x Nix 24 glasses Detailed kinetics studies on the crystallization process in binary Zr–Fe, Zr– Ni and ternary Zr–Fe–Ni glasses have revealed the characteristics of different modes of crystallization. A number of investigations (Buschow 1981, Dey and Banerjee 1985a,b, Dey et al. 1986, Ghosh et al. 1991) have established that while the Zr 76 Fe24 glass crystallizes polymorphically to the Zr 3 Fe phase (body centred orthorhombic, Re3 B type structure), the Zr 76 Ni24 glass decomposes into a mixture of the hcp and the Zr 2 Ni (body centred tetragonal) phases on crystallization. As Ni substitutes for Fe in ternary alloys, Zr 76 Fe1−x Nix 24 , (x = 0 4 8 12 16 20 24), the crystallization products are expected to change as the equilibrium Zr 3 Fe and Zr 2 Ni phases have limited solubilities in respect of Ni and Fe, respectively. The partitioning of Ni and Fe atoms among the growing crystalline phases during crystallization is also expected to have a significant influence on the kinetics and mechanisms of the crystallization process. The work of Dey et al. (1998) has demonstrated how ternary additions influence the crystallization behaviour of Zr 76 Fe1−x Nix 24 alloys. In view of the fact that the common modes, namely polymorphic, eutectic, primary followed by eutectic and surface crystallization, are all encountered in this system, the results of this work are summarized here for highlighting the different kinetic features which characterize these modes of crystallization. The number and the nature of the thermal events accompanying crystallisation in these glasses are shown in DSC thermograms (Figure 3.38(a)), all of which correspond to a constant heating rate of 10 K/min. The glass transition temperature, Tg , the crystallization start temperature, Tx , and the peak transformation rate temperature, Tp , are listed along with the sequence and the mode of crystallization in Table 3.6. The progress of crystallization, as detected during isothermal holding in a DSC, in all these alloys is shown in Figure 3.38(b). Some special features of these exotherms need special mention: (a) the presence of a well-defined and sufficiently long incubation period prior to crystallization in glasses of the following compositions: Zr 76 Fe24 , Zr 76 Fe20 Ni4 , Zr 76 Fe16 Ni8 , Zr 76 Fe12 Ni12 and Zr 76 Ni24 ; (b) very short incubation periods for the glasses Zr 76 Fe8 Ni16 and Zr 76 Fe4 Ni20 ; and (c) alloys grouped in (a) show symmetric exotherms while those grouped in (b) exhibit very broad asymmetric exotherms. Phase analysis by XRD and electron microscopy and diffraction on specimens crystallized at temperatures in the range of 573–673 K for different durations have provided information regarding the identity of the crystalline phases formed and the mode of crystallization as well as the nucleation density and growth rate of crystalline particles as functions of temperature and time. These are complementary to kinetics data obtained from the bulk samples. The occurrence of surface crystallization could be detected from metallographic examinations of cross-sections

Rate of heat flow


201

1 3 2 4

5

6 7

600

620

640

660

680

700

720

Temperature (K)

Rate of heat flow

(a)

631 K 7

4

631 K

5 6

00

05

10

3

2

620 K 15

632 K

1

633 K 630 K

632 K 20

25

30

35

40

45

50

Time (min)

(b)

Figure 3.38. (a) DSC thermogram showing the number and the nature of the thermal events accompanying crystallization in Zr 76 Fe1−x Nix 24 glasses at compositions (i) Zr 76 Fe24 , (ii) Zr 76 Fe20 Ni4 , (iii) Zr 76 Fe16 Ni8 , (iv) Zr 76 Fe12 Ni12 (v) Zr 76 Ni24 , (vi) Zr 76 Fe8 Ni16 and (vii) Zr 76 Fe4 Ni20 ; (b) DSC thermogram showing the progress of crystallization during isothermal holding Zr 76 Fe1−x Nix 24 glasses.

of partially crystallized specimens and by carrying out XRD of specimens before and after the removal of surface layers of about 10 m thickness. Taking into account the results obtained from microscopy and DSC experiments, the following inferences could be drawn. DSC thermograms for the alloys Zr 76 Fe8 Ni16 and Zr 76 Fe4 Ni20 , which do not show long incubation periods, are associated with the surface crystallization process which precedes bulk crystallization. The product phase resulting from surface crystallization has been identified to be an ordered cubic phase with the L12 structure (lattice parameter: 0.420 nm) and an approximate composition of Zr 3 (Fe,Ni). The exotherms for these two glasses show a limited overlap of two peaks – the first

202


one being asymmetric, characteristic of the primary crystallization of Zr3 (Fe,Ni) occurring on the surface and the second symmetric peak corresponding to the predominant eutectic crystallization occurring in the bulk. The J–M–A analysis of the kinetic data which requires a deconvolution of the two peaks yields the Avrami exponents, n1 and n2 , for the two processes. The Zr 76 Fe24 glass crystallizes polymorphically to yield the equilibrium Zr 3 Fe phase. The observed Avrami exponent of 3 corresponds to an interface-controlled growth of a spherical transformed product (Zr 3 Fe crystals in the present case). As mentioned earlier, polymorphic crystallization is a composition invariant process in which the kinetics are controlled by the mechanism of short-range atom transport across the crystal/amorphous matrix interfaces. The activation energy obtained from the J–M–A analysis of the isothermal kinetics corresponds to the overall kinetics including both nucleation and growth; activation energies for these steps can be determined separately by measurement of the number density and the size of largest crystals formed in specimens annealed at different temperatures for different durations. TEM examinations of samples which have undergone different extents of crystallization provide such informations (plotted in Figure 3.36 for both the Zr 76 Fe24 and the Zr 76 Ni24 glasses). The Zr 76 Ni24 glass crystallizes in an eutectic mode to produce a mixture of the

-Zr and the Zr 2 Ni phases. Nodules of this two-phase mixture nucleate and grow to consume more and more of the amorphous matrix. No long-range diffusion field is created around these growing nodules. As a consequence, these nodules grow until they come in contact with adjacent nodules. No amorphous matrix is retained at this stage. The observed Avrami exponent, n = 32, is again consistent with an interface-controlled growth of crystalline aggregates in three dimensions. Though there is a partitioning of solutes at the interface between the amorphous matrix and the two product phases, the absence of a long-range diffusion field ahead of the growing particle makes the process interface controlled. Figure 3.36(a) and (b) shows that the rates of nucleation and growth remain constant (linear growth) with time for both polymorphic (in the case of Zr 76 Fe24 ) and eutectic (in the case of Zr 76 Ni24 ) crystallization. For such cases, the temperature dependence of the growth rate, dr/dt, can be expressed as dr/dt = exp−Qg /RT 1 − exp− G/RT

(3.54)

where r is the radius of the growing crystal at time t, G is the change in the chemical free energy per mole accompanying crystallization, Qg is the activation energy for growth, is the characteristic frequency and is the interface width. Since crystallization experiments are carried out at temperatures where glasses are supercooled to a great extent, G 0 V V TCi

(3.58)

where , ! and are shear stress, strain and modulus, respectively, and B is the bulk modulus. The elastic constants of a superheated crystal, (= C11 − C12 /2, isothermal shear modulus), and B (= C11 + 2C12 /3, isothermal bulk modulus), vanish at the critical temperatures, T and TB , respectively, which are above the melting temperature, Tm . Experimental data on elastic moduli as function of temperature (Tallon and Wolfenden 1979), when extrapolated to temperatures above Tm , show that the instability sets in at T ∼ 16Tm (Figure 3.48). The increase in thermal disorder as the temperature is raised leads to an ultimate shear instability of a metastable superheated crystalline solid. Other types of disorder, namely defects and chemical disorder, can also contribute in bringing about the shear instability. Thorpe (1983) has theoretically studied the mechanical stability of a model crystal, consisting of balls of equal mass M connected with nearest neighbour balls by springs with force constant K. Beginning with the perfect crystal, the springs are gradually removed from random locations. Using computer simulation, the shear and bulk moduli are calculated as functions of the fraction of springs removed. The results of this theoretical work have shown that both and B vanish where a critical fraction of springs are removed (Figure 3.49).

216


8

Tm

Elastic moduli 1010 Pa

B 6

4

μ

2

200

Instability

600

1000

1400

1800

Temperature

Elastic moduli (arb. unit)

Figure 3.48. Variation of elastic moduli as a function of temperature.

B

μ

Fraction of removed springs

Figure 3.49. Shear and bulk modulus as function of fraction of springs removed.

Egami and Waseda (1984) have considered a binary solid solution containing atoms of two different sizes. By carrying out a simple analysis of local strain effects using an elastic continuum approach, they have shown that such a solution becomes topologically unstable when the concentration of the smaller atoms (A atoms)


217

reaches a critical concentration, cA∗ , which depends on the ratio of atomic sizes r = RA /RB , where RA and RB are the atomic radii of the two types of atoms. Considering the instability arising out of a critical level of strain disorder, the concentration level at which the solid solution becomes topologically unstable has been found to be cA∗ = 2R3B /R3B − R3A + higher order terms

(3.59)

Let us now examine the free energy – concentration diagram of a system at a temperature below Tg (for the alloy compositions under consideration). Figure 3.50 shows such a diagram which depicts the condition of metastable equilibrium between the crystalline -phase and the amorphous phase (a). The presence of the equilibrium intermetallic phase is denoted here with a dotted line. If the formation of the intermetallic phase is kinetically prevented, nucleation of the amorphous phase becomes thermodynamically possible only when the concentration of B atoms in the -phase, cB exceeds the limit cB1 . The maximum free energy change associated with the nucleation of the amorphous phase from an -phase having a composition given by cB is shown by the vertical line Gm . Such a nucleation process, which is facilitated at heterogeneities like high-energy grain boundaries, involves the partitioning of B atoms preferentially towards the nucleating amorphous phase. If, however, the -phase is enriched to a composition where cB > co , a massive (i.e. partitionless) → a transformation becomes possible.

α

Free energy

a

ΔG m

α

c1B

cB co c B2

A

CB

B

Figure 3.50. Free energy–composition diagram showing metastable equilibrium between the crystalline -phase and the amorphous phase.

218


As discussed earlier in Section 3.4.1, the phase diagrams of typical glass forming alloys are characterized by steeply plunging To -lines, as seen for Zr–Ni and Zr–Cu alloys. For such alloys, a generic non-equilibrium phase diagram can be developed, neglecting the kinetically excluded intermetallic compounds, for illustrating the possible thermodynamic states of a metastable system constrained to be a single phase. For alloys with large negative slopes for the To lines, the To line must cross the ideal glass transition line Tg∗ at a certain composition, c∗ . Under this condition, a triple point (c∗ , T ∗ ) is defined in respect of the supersaturated crystal, undercooled liquid and ideal glass (Figure 3.51). The entropic instability line, Tis , against melting (shown by broken line) should pass through the triple point since the entropy difference between the crystal and the liquid vanishes at this point. This polymorphous phase diagram (Figure 3.51) essentially depicts that the following two conditions are satisfied: G = H − T S = 0 (condition for polymorphic melting) and S = 0 (Kauzman condition) at the triple point. This also shows that the composition-induced disorder reduces the polymorphic melting temperature of the crystalline solid solution to the ideal glass transition temperature. The slope of the To line, dTo /dc = − G/c/ S at the triple point approaches infinity. Fecht et al. (1989) have predicted the triple point for -Zr supersaturated with Ni to be 638 K and 11.5 at.% Ni. The To line extends below the triple point as a

Temperature

T

s i

To

Liquid

T g* Crystal

T* C*

Glass

Composition

Figure 3.51. Schematic diagram indicating a triple point (c∗ , T ∗ ) as defined in respect of the supersaturated crystal, undercooled liquid and ideal glass.


219

straight line with infinite slope as long as non-ergodicity prevails and the Kauzman argument holds. Below the triple point, the transition between the crystal and the glass is isentropic and, therefore, truly continuous in volume as long as the metastable constraints are maintained. The presence of non-equilibrium lattice defects such as vacancies and anti-site defects play a major role in providing the constraint under which the melting temperature of the crystal can be considerably reduced. These defects, which have very low mobility at relatively low processing temperatures, remain frozen in the lattice. If one considers the vacancy as a second component in the system, one can draw a phase diagram like the one shown in Figure 3.52. The free energy difference, G = H − T S, between a liquid and a single crystal for pure metals can be realistically estimated to be (Fecht et al. 1989)

G = 7 Sf TTTm + 6T

(3.60)

and for glass forming alloys by

G = 2 Sf TTTm + T

(3.61)

where T is the undercooling below the melting point, Tm , and Sf the entropy of fusion. The increase in free energy of the crystalline phase can be expressed as

Gv = cv H v − T S v + kB T cv ln cv + 1 − cv ln1 − cv

T/ Tm

1.0

(3.62)

ΔG = 0 Liquid

0.8

Crystal 0.6

T∗

0.4

ΔS = 0

Glass

0.2

c °v

ΔH = 0 0

0.02 0.02

0.04

0.06

0.08

0.10

Vacancy concentration (c v)

Figure 3.52. Phase diagram illustrating the role of non-equilibrium lattice defects such as vacancies and anti-site defects in providing the constraint under which the melting temperature of the crystal can be considerably reduced.

220


Combining these equations, the decrease in melting temperature of the defective crystal can be expressed as a function of defect concentration. 3.7.2 Amorphous phase formation by composition-induced destabilization of crystalline phases There are a number of experimental results to demonstrate that a crystalline material can be transformed into an amorphous one by progressively introducing alloying elements. It has often been noticed that a crystalline phase is destabilized when loaded with some specific alloying elements to a level exceeding a certain threshold. Introduction of alloying elements can be effected by several means, for example, (a) by isothermal annealing of diffusion couples, (b) by mechanical alloying, (c) by introducing hydrogen by diffusion and (d) by ion implantation. All these treatments, which are essentially isothermal but are implemented under chemically non-equilibrium conditions, can lead to the formation of amorphous phases. It is the excess chemical energy associated with the initial configuration which permits the glassy state to be adopted and retained as a metastable product. Some experimental results pertaining to the aforementioned treatments will now be described, and glass formation will be rationalized in terms of thermodynamics and kinetics of the pertinent process. 3.7.3 Glass formation in diffusion couples The early observations on glass formation in diffusion couples were reported in initially crystalline multilayers of Au–La (Schwartz and Johnson 1983) and in samples of Si coated with a thin film of Rh (Herd et al. 1983). The formation of an amorphous layer in the reaction zone of binary diffusion couples has been observed in a number of systems in which one of the components is Zr or Ti. These are the well-known glass forming systems such as Zr–Cu, Zr–Ni, Zr–Co, Zr–Fe and Ti–Ni. A variety of techniques have been employed for detecting the amorphous phase. Serial sectioning of diffusion couples near the reaction zone provides samples for a plan view examination by XRD and TEM, while cross-sectional TEM reveals the presence and distribution of different layers forming at the reaction zone. The composition profile is determined by using electron proble microanalysis and Rutherford backscattering. The presence of the amorphous phase can also be detected by the observation of a crystallization event during heating in a DSC. Let us examine some of the reported experimental results in order to understand the thermodynamics and kinetics of the formation of amorphous layers in the reaction zones of diffusion couples. Cross-sectional TEM studies on Zr–Ni diffusion couples by Newcomb and Tu (1986) have shown the presence of a well-defined planar interlayer of an amorphous phase in diffusion couples reacted at 573 K for durations of 1.5 and 4 h. The formation of the ordered intermetallic ZrNi phase has


221

Kirkendal voids

Amorphous interlayer

NiZr

20000

Zr

–10000 –20000

= Equilibrium compounds

550 K

10000

ΔG (kJ/mol)

Ni

HEX FCC

BCC

–30000 –40000

Amorphous –50000 –60000

Ni

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Zr

cZr

(a)

(b)

Figure 3.53. (a) Schematic diagram showing the distribution of different phases and of Kirkendal pores in a Zr–Ni diffusion couple reacted at 573 K for 12 h, (b) free energy concentration diagram in Zr–Ni system.

been observed when the reaction time is extended to 12 h. A schematic illustration (Figure 3.53(a)) shows the distribution of different phases and of Kirkendal pores in a Zr–Ni diffusion couple reacted at 573 K for 12 h (Newcomb and Tu 1986). Free energy–concentration plots for the different competing phases in this system (Figure 3.53(b)) can be used for illustrating two possible mechanisms for the formation of the amorphous layer. The normal downhill interdiffusion process is expected to produce a hcp solid solution of Ni in Zr and an fcc solid solution of Zr in Ni. The amorphous phase can either nucleate heterogeneously at the interface, the local metastable equilibrium being dictated by tangent construction on the free energy curve for the terminal solid solution or by solute enrichment of the solid solution to an extent where it reaches beyond the co limit (when the system is forced below the To temperature corresponding to the composition, co ) leading to polymorphic vitrification. In order to find out whether a solid solution phase in the Ni–Zr system can become unstable with respect to the Gibbs criterion, it is necessary to determine the minimum level of solute content which allows a massive or polymorphic vitrification. For the solution of Ni in hcp Zr, the To cN i ) line reaches 0 K at cNi = 022. The criterion based on the local strain, as proposed by Egami (1983), gives the composition limits for instability as cNi = 026 and cZr = 013 for the hcp and fcc solid solutions, respectively. From these estimations of instability limits, it appears possible that the destabilization of a solid solution phase can indeed occur, leading to a polymorphic vitrification process. It should, however,

222


be noted that the nucleation of an amorphous phase by heterogeneous nucleation becomes thermodynamically feasible at much lower levels of the solute content in terminal solid solutions. The importance of the heterogeneous nucleation of an amorphous phase in diffusion couples can also be realized from the observation that an amorphous phase does not form in a couple made by depositing Ni on a single crystal of Zr. The question of polymorphic vitrification vis-a-vis nucleation of an amorphous phase of a composition given by the local metastable equilibrium condition has been addressed by Bhanumurthy et al. (1988, 1989) while analysing the results pertaining to Zr–Cu diffusion couples. These experiments have shown that an interface reaction between Zr and Cu at a temperature of 873 K results in the formation of an ordered intermetallic phase, Zr2 Cu, whereas at a temperature close to 600 K, an amorphous layer forms in the reaction zone. At such low temperatures, the formation of the ordered intermetallic phase is fully suppressed. An examination of hypothetical free energy – concentration plots for the , the and the liquid phases (Figure 3.54) reveals that the metastable solubility of Cu in the -Zr lattice is considerably extended when intermetallic phase formation is suppressed. The equilibrium conditions between the different competing phases are shown by the common tangents AB (between hcp and Zr2 Cu), CD (between

and the amorphous a-phase), EF (between and the bcc -phase) and finally JK (between and L). The points of intersection in the free energy – concentration

T~600 K

H Ga

Free energy (G)

Gα

G Zr2Cu

IJ F

A C E

Gβ M

N T

AB: G α, G Zr2Cu

K

CD: G α, G a

D P

α

ET: G at E EF: G α, G β

B

JK: G β, G a 0.1

0.2

0.3

0.4

0.5

Atomic fraction (Cu)

Figure 3.54. Free energy – concentration plots for , , Zr2 Cu and the liquid phases.


223

curves for different phases mark the limits at which polymorphic transformations between them become thermodynamically possible. For the -phase to become amenable to polymorphic vitrification, the Cu enrichment of this -phase should go beyond the point H. Before attaining such a high Cu concentration, the supersaturated -phase becomes amenable to a composition invariant massive to transition, which has not been observed in the reaction zone of Zr–Cu diffusion couples. Microanalysis of the different phases present in the reaction zone has shown that the maximum concentration of Cu in the -phase, which lies in contact with the amorphous layer, is about 4 at.%. Based on these observations, it has been inferred that the formation of the amorphous phase in the reaction zone of Zr–Cu diffusion couples occurs by its nucleation from the -phase, enriched to a Cu content of about 4 at.% and its subsequent growth by the coalescence of independently nucleated amorphous regions. Once the -phase is enriched in Cu to a level shown by the point E, the maximum free energy change for the nucleation of the amorphous phase is given by the drop NP where P is given by a tangent drawn on the Ga plot which is parallel to ET. A continuous amorphous layer eventually develops which grows in thickness by consuming the adjacent -phase layer till the supply of Cu atoms gets restricted by the formation of Kirkendal pores on the Cu-rich side. The Cu level required for the destabilization of the -Zr lattice, on the basis of the Egami criterion, has been estimated to be about 28 at.%. This lends further support to the contention that the amorphous layer forms in the Zr–Cu system not by the destabilization of the -lattice but by the nucleation of a Cu-rich amorphous phase which can establish a metastable equilibrium with an

-Zr–Cu alloy containing about 4 at.% Cu. The kinetics of the one-dimensional growth of the amorphous interlayer in the reaction zone of a binary diffusion couple can be described by a set of coupled differential equations (Johnson 1986): c 2 c ˜ =D t x2 ˜ D ˜ −D

(3.63)

dx c x1 = 1 − c1 1 x dt

(3.64)

dx c x2 = c2 2 x dt

(3.65)

dx2 = K2 c2 − c2o dt

(3.66)

dx1 = K1 c1 − c1o dt

(3.67)

224

Phase Transformations: Titanium and Zirconium Alloys dX2 dt

dX1 dt 1

GLASS

2

X

1.0

c10 c 20

c1 0 K1 ~ D

μ1

K2

μ2

X1

X2

Figure 3.55. Schematic diagram showing the concentration profile of metal 1 and chemical potential profile of metal 1 and metal 2.

˜ is the interdiffusion constant in the amorphous phase, c(x) is the concenwhere D tration profile of metal 1 in the amorphous phase, c1o and c2o are the concentrations of metals 1 and 2, respectively, in the amorphous phase which are in equilibrium with pure metal 1 and pure metal 2 (as shown in Figure 3.55); x1 and x2 give the positions of the interfaces separating the amorphous layer and the metals 1 and 2, K1 and K2 are the kinetic response parameters at these interfaces and c1 and c2 are abbreviated forms of cx1 and cx2 . ¯ Introducing some simplifying assumptions such as the interdiffusion constant D being independent of composition and the response parameters K1 and K2 being linear and being related as K1 /K2 = c1o /1 − c2o

(3.68)

The equations have been solved numerically and for long times (t → ), ˜ 2 + 2aDt ˜ x2 = −D/K

(3.69)


225

where a is a constant of order unity. For short times (t → 0) the solution is in the following form: x2 = constantK2 t + negligible higher order terms

(3.70)

These results predict a linear growth law at short times (for a thin amorphous interlayer) and a shifted t1/2 law in the limit of long times. This means that when ¯ the amorphous layer is thinner than the characteristic length, l= D/K, the growth is interface controlled while a diffusion controlled growth mechanism operates when the amorphous interlayer is much thicker than l. Analysis of experimental data on the growth of amorphous interlayers in diffu˜ the interdiffusion constant, sion couples of Ni–Zr, Co–Zr and Ni–Hf has yielded D, ˜ values as a function of temperature at T < Tg . Johnson (1986) has shown that D match very closely with the diffusion constant for impurity diffusion of Ni in the amorphous Ni67 Zr33 alloy. This observation points to the fact that the interdiffusion process is strongly dominated by the migration of the smaller atoms of the late transition metals (Ni in the case of Zr–Ni) and practically no migration of Zr atoms. The formation of Kirkendal voids along the interface separating Ni from the amorphous layer is a direct evidence that Ni is the moving species. The void formation is responsible for reducing interfacial contact and ultimately for cutting off the supply of Ni atoms. At this stage, the growth of the amorphous layer terminates. Experiments have shown that the growth of amorphous interlayers can lead to a layer thickness of 100–200 nm without any accompanying formation of crystalline intermetallic compounds. Since this thickness is much larger than the size of critical crystalline nuclei, the avoidance of crystalline nucleus formation during interdiffusion annealing for a time scale of about 104 s appears improbable. The fact that only the late transition metal atoms are mobile with virtually no migration of Zr atoms can perhaps explain why nucleation of ordered intermetallic compounds does not occur during the growth of the amorphous interlayer.

3.7.4 Amorphization by hydrogen charging Yeh et al. (1983) have reported that a bulk polycrystalline partially ordered fcc solid solution of Zr075 Rh025 composition transforms to an amorphous phase when hydrided by exposure to hydrogen gas at temperatures between 425 and 500 K. This reaction can be expressed as Zr 075 Rh025 cryst + H2 gasZr 075 Rh025 H114 amorphousT < 500 K

(3.71)

226


At higher temperatures (T > 500 K) the equilibrium product forms as per the following equation: Zr 075 Rh025 cryst + H2 gasZrH2 cryst + RhcrystT > 500 K

(3.72)

TEM observations have revealed that at T < 500 K, amorphization proceeds by the nucleation of glassy zones along the grain boundaries of the crystalline starting material, followed by the expansion of these amorphous regions into the grain interiors. The boundary between the crystalline and the amorphous regions is sharp, suggesting a strong first-order transition. The entire process proceeds much as melting would proceed in a polycrystalline sample. X-ray studies have indicated that the fcc Zr075 Rh025 solid solution dissolves some hydrogen prior to transforming to the glassy phase. This suggests that with hydrogen entry, the free energy of the alloy is raised above that of the glassy phase. As a consequence, the superheated crystalline phase transforms into an amorphous phase. The temperature threshold, 500 K, below which amorphization occurs, is dictated by the relative values of the diffusion constants of H and the metal atoms. Above 500 K the equilibrium product consists of two crystalline phases, namely fcc Rh and fcc ZrH2 . For the formation of such a product, metal atom redistribution must occur by thermally activated diffusion over a length scale at least of the order of the respective critical nuclei sizes of the two crystalline phases. This is apparently not possible below 500 K. The temperature dependence of the chemical rate constants of Eqs. 3.71 and 3.72 is expected to be considerably different as these processes are controlled by hydrogen and metal atom diffusion, respectively. It is this difference in temperature dependence that enforces a kinetic constraint on the separation of the two crystalline phases and instead allows the hydrogen-charged Zr75 Rh25 alloy to undergo amorphization in a manner similar to melting. 3.7.5 Glass formation in mechanically driven systems High energy ball milling can lead to glass formation from elemental powder mixtures as well as by amorphization of intermetallic compound powders. Solid state amorphization by high energy milling has been demonstrated in a number of Ti- and Zr-based and other alloy systems such as Ni–Ti, Cu–Ti, Al–Ge–Nb, Sn– Nb, Ni–Zr, Cu–Zr, Co–Zr and Fe–Zr. The process of ball milling is illustrated in Figure 3.56. Powder particles are severely deformed, fractured and mutually cold welded during collisions of the balls. The repeated fracturing and cold welding of powder particles result in the formation of a layered structure in which the layer thickness keeps decreasing with milling time. A part of the mechanical energy accumulates within these powder particles in the form of excess lattice defects which facilitate interdiffusion between the layers. The continuous reduction in the diffusion distance and the enhancement in the diffusivity with increasing milling


227

A Powder particle B

Hard ball

Figure 3.56. Schematic diagram illustrating the process of ball milling.

time tend to bring about chemical homogeneity of the powder particles by enriching each layer with the other species being milled together. The sequence of the events that occur during milling can be followed by taking out samples from the ball mill at several intervals and by analysing these powder samples in respect of their chemical composition and structure. Let us describe one such experiment in which elemental powders of Zr and Al were milled in an attritor under an Ar atmosphere. Elemental powders of Zr and Al of 99.5 purity, when milled in an attritor using 5 mm diameter balls of zirconia as the milling media and keeping the ball to powder weight ratio at 10:1, showed a progressive structural change as revealed in XRD patterns (Figure 3.57(a) and (b)). Diffraction peaks associated with the individual elemental species remained distinct upto 5 h of milling at a constant milling speed of 550 rpm. All particles and the balls appeared very shiny in the initial stages. With increasing milling time, the particles lost their lustre, the 111 and 200 peaks of fcc Al gradually shrunk and the three adjacent low-angle ¯ 0002 and 1011, ¯ became broader. After peaks of hcp -Zr, corresponding to 1010, about 15 h of milling, XRD showed only -Zr peaks which shifted towards the high angle side, implying a decrease in the lattice parameters resulting from the enrichment of the -Zr phase with Al. After 20 h of milling, all Bragg peaks except one broad peak close to the {1010} peak disappeared. Powders milled for 25 h showed an extra reflection corresponding to a lattice spacing of 5.4 nm, which matches closely to a superlattice reflection of a metastable D019 (Zr3 Al) phase. On further milling, the powders transformed into an amorphous phase. The sequence of structural evolution could be described as -Zr + Al −→ -Zr (Al) solid solution + Al −→ nanocrystalline solid solution + localized amorphous phase −→ Zr3 Al (D019 ) + -Zr (Al) solid solution + amorphous phase −→ bulk amorphous phase.


Zr (002)

Zr (101) a–5 h b – 10 h c – 15 h d – 20 h

a – 25 h b – 30 h c – 45 h

Intensity (A.U.)

Zr (100) Al (111) Zr (102) Al (200)

a

Intensity (A.U.)

228

b c d 22

26

30

34

38

42

46

50

16

20

24

28

32

2–θ

2–θ

(a)

(b)

36

40

44

48

Figure 3.57. XRD patterns showing a progressive structural change for different times when elemental powders of Zr and Al of 99.5 purity were milled in an attritor using 5 mm diameter balls of zirconia with a ball to powder weight ratio of 10:1.

The mechanism of solid state amorphization during mechanical alloying has been studied on the basis of experimental observations made on several alloy systems. One of the probable mechanisms, based on local melting followed by rapid solidification, has not found acceptance as evidence of melting could not be seen in experiments. The example of ball milling of elemental Zr and Al powders has demonstrated that the amorphisation process is preceded by the enrichment of the -Zr phase to a level of approximately 15 at.% Al. The solute concentration progressively changes during milling. The various stages encountered in the course of amorphization can be explained in terms of schematic free energy versus concentration plots for the , the metastable D019 , and the amorphous phases (Figure 3.58). With increasing degrees of Al enrichment, the free energy of the interface region gradually moves along the path 1-2 (Figure 3.58). Once the concentration crosses the point 2, it becomes thermodynamically feasible to nucleate the Zr3 Al phase which has the metastable D019 structure. Although the equilibrium Zr3 Al phase has the L12 structure, it has been shown (Mukhopadhyay et al. 1979) that the metastable D019 structure is kinetically favoured during the early stages of precipitation from the -phase. This is not unexpected as the hcp


229

Zr3 Al (DO19)

α

1

Amorphous

Free energy

2 3

4

3′

2′

Zr

25 Aluminium content (at.%) →

Figure 3.58. Schematic free energy – concentration plots in Zr–Al system for the , the metastable D019 and the amorphous phases illustrating the various stages encountered in the course of amorphization.

structure and the D019 structure (which is an ordered derivative of the former) follow a one-to-one lattice correspondence and exhibit perfect lattice registry. With further Al enrichment, as the concentration crosses the point 3, nucleation of the amorphous phase becomes possible. It is to be emphasized that the change in composition occurs gradually from the interface to the core of the particles, with the result that the amorphous phase starts appearing at interfaces while the core remains crystalline. As the Al concentration in the powder particles crosses point 4, each particle can turn amorphous by a polymorphic process. The observed sequence of solid state amorphization in the case of ball milling of elemental Zr and Al powders suggests the occurrence of amorphization by a lattice instability mechanism which is brought about by solute enrichment of the -phase beyond a certain limit (point 4 in Figure 3.58). 3.7.6 Radiation-induced amorphization It was discovered in the early 1980s that some intermetallic compounds undergo a crystalline to amorphous (C −→ A) transition under irradiation by energetic particles. It was also recognized that the C −→ A transition results from displacement damage and not from ionization damage. Displacement damage occurs due to the momentum transfer from the projectile particles (incident electron, ion or neutron) to atoms occupying lattice sites in the target material. For example, if one considers an elastic collision between an incident particle of mass m and an atom of atomic weight M, the struck atom can receive the maximum energy,

230


Emax = 4Mm/En M + m2 , where En is the energy of the incident particle. While an 1 MeV neutron can transfer several hundred keV energy to a target atom, a lighter particle such as an electron of the same energy (1 MeV) is capable of transferring only some tens of eV energy to a target atom. The threshold energy, Ed , for displacing an atom from its lattice site is in the range of 20–80 eV ( 25 eV for Cu) and so an electron of 1 MeV energy can displace only one or two atoms from lattice sites. In contrast, a 1 MeV neutron can impart an energy of as much as 200 000 eV to a Cu atom. Such an atom, called a primary “knock-on”, causes further damage by displacing secondary, tertiary, etc. “knock on”s. The damage structure produced by a single energetic electron will, therefore, consist of a single (or two) vacancy – interstitial pair, the separation between the vacancy and the interstitial being dictated by the length of the replacement collision sequence. Figure 3.59(a) and (b) shows schematically the defect production process due to an incident energetic electron. In contrast, a high-energy neutron (or an accelerated ion) produces a row of primary “knock-on”s, each of which triggers a series of displacements in its path, as shown in Figure 3.59(c). The cascade of displacements finally terminates where the energy transferred to the target atoms falls below the threshold energy for atomic displacement. At these termination points, the displaced atoms deposit their energies by thermal excitation of neighbouring atoms. Such a thermal excitation has a very short life span (10−12 s) and is known e

e V

V I

V

I

I (a)

(b)

Ion

(c)

Figure 3.59. Schematic diagrams showing the defect production process due to an incident energetic electron ((a) and (b)). In contrast, a high-energy neutron (or an accelerated ion) produces a row of primary “knock-on”s, each of which triggers a series of displacements in its path (c).


231

as a thermal spike. The region over which structural change occurs due to a cascade of displacements is known as a “displacement cascade”, the size of which is determined by the mass and the energy of the projectile particle and the mass and the threshold displacement energy of the target atoms. Since a large number of atoms are expelled from the core of the cascade, this depleted region contains a high density of vacancies while the periphery of the cascade gets enriched in interstitials. Often interstitials are produced at the end of a replacement collision sequence chain which propagate along close packed directions from the core to the periphery of the cascade. The brief description of the radiation damage processes, pertinent to irradiation by electrons and relatively heavy particles (such as neutrons and ions), given here provides a background for gaining an understanding of why and how radiation damage induces the crystal to amorphous transformation primarily in intermetallics. It has been observed that some compounds undergo amorphization while others remain crystalline under similar irradiation conditions. The observed difference in the susceptibility to amorphization under irradiation has led to the identification of several empirical criteria which promote amorphization. (1) Directional bonding such as ionic and covalent bonding is a requirement as evidenced from the fact that pure metals (with the exception of Ga) and disordered solid solutions cannot be amorphized under irradiation (Cahn and Johnson 1986). The melting point of Ga at ambient pressure is anomalously low (Tm = 302 K, the heat of fusion being 0.6 kcal/mol). This corresponds to a very small difference in the free energies of the crystalline and amorphous states of Ga at low temperatures. The small requirement of enthalpy for the crystal to amorphous transition can be met by the energy stored in the form of point defects which are produced under irradiation. Semimetals such as Si, Ge and Bi are also amenable to amorphization under irradiation – an observation which is consistent with this criterion. (2) Intermetallic compounds which exhibit narrow solubility ranges in the phase diagram tend to amorphize under irradiation, while those with wide solubility ranges remain crystalline. Though this criterion is not universal, one can argue that a narrow solubility range of composition corresponds to a high energy being associated with the anti-site defects created during irradiation. A large energy storage through these defects may eventually lead to amorphization. (3) The presence of deep eutectics and of a number of line compounds in the same region of a phase diagram is indicative of a relatively high stability of the liquid phase and of an inclination towards chemical ordering in the system. These thermodynamic features are usually associated with a tendency for amorphization not only under irradiation but also during rapid solidification (see Section 3.4). Binary alloys with constituents from the early and the late transition metals such

232


as Zr–Ni, Zr–Fe, Zr–Cu, Ti–Ni and, Ti–Cu exhibit phase diagrams which satisfy this criterion and are known to be amenable to amorphisation under irradiation. Since the amorphous state is metastable with respect to the unirradiated crystalline state, the occurrence of a crystal to amorphous (C → A) transformation under irradiation is possible only if the increase in free energy, Girr , due to irradiation is greater than the difference between the free energies, GC→A , of the crystalline and amorphous phases, i.e.

Girr > GC→A

(3.73)

Some of the energy input from the incident radiation must, therefore, be stored permanently (or for a time comparable to the amorphization time) in the material. Since most of the irradiation energy is dissipated as heat, a question arises as to the mechanism by which enough energy could be accumulated in the lattice for fulfilling the above criterion. There are two important mechanisms of energy storage within an irradiated material: first, by accumulation of point defects, both vacancies and interstitials, much in excess of their equilibrium concentrations, and second, by the creation of anti-site defects produced by random displacements and by replacive collision sequences. Let us now examine the relative merits of these two conceivable energy storage mechanisms. One can estimate the critical vacancy concentration required to raise the enthalpy of a metallic crystal at T = 0 K by an amount equal to the heat of fusion, Hf , at the melting temperature, Tm . Doyama and Koehler (1976) have shown that the critical vacancy concentration, cv , required for raising the free energy of a crystal at T = 0 K to the free energy of an amorphous phase is given by cv = Hv / Hf 0008

(3.74)

where Hv is the enthalpy of formation of a vacancy. The question that now arises is whether the steady state vacancy concentration in a crystal can be raised to a level as high as 8 ×10−3 under irradiation. The steady state vacancy concentration under irradiation depends on the rate of production of vacancy – interstitial pairs, the rate at which they recombine and the rate of disappearance of vacancies at sinks such as surfaces, interfaces and dislocation loops. It has been found that even at low temperatures and at high displacement rates (∼103 displacements per atom), the steady state vacancy concentration does not exceed about 10−3 . This is primarily because of the high probability of recombination of vacancies and interstitials as their concentrations increase with the displacement rate. Even when the enthalpy of formation of interstitials (whose steady state concentration remains at one or two orders of magnitude lower than


233

that of vacancies) is added to that of vacancies, the free energy of the crystalline phase of a pure metal cannot be raised above the free energy of its amorphous phase by introducing point defects to a realistic maximum concentration. The second mechanism of storing energy in the crystal lattice exposed to irradiation is by the creation of anti-site defects. Obviously such a mechanism can operate only in chemically ordered structures. Let us consider the example of the B2 (CsCl type) ordered structure. Here the lattice can be viewed as comprising two interpenetrating simple cubic sublattices, denoted as and . In the fully ordered condition, the -sites are occupied by only A atoms and the -sites are occupied by only B atoms. Each -site is surrounded by eight nearest neighbour -sites and vice versa. In such a structure, vacancies in - and - (denoted by and ) sites can be distinguished and their exchanges with atoms lead to a change in the order parameter, S. Under irradiation, a high steady state vacancy concentration promotes atom– vacancy exchange process, as shown below: A + v A + v

(3.75)

B + v B + v

(3.76)

A + B A + B

(3.77)

Equation 3.77 shows the exchange of A and B atoms from the to -sites and vice versa, leading to disordering in the forward reaction and to ordering in the backward reaction. The overall reaction, however, involves two steps corresponding to two successive atom–vacancy site exchanges as shown in Eqs. 3.75 and 3.76. The energy barriers for the reactions of A and B atoms with vacancies on the and sublattices in a fully ordered (order parameter, S = 1) and a fully disordered (S = 0) alloy are shown in Figure 3.60. Four different types of atom– vacancy interchange processes are indicated. In a completely disordered alloy, such exchanges do not lead to any change in the energy of the system and therefore in the activation barrier, Em , since such jumps are symmetric. In the partially or fully ordered alloy sublattice, changes of vacancies contribute to an alteration in the degree of order and lead to a decrease or increase in the energy of the system by an amount U . By linear interpolation, one obtains for the contribution at the saddle point, one half of this energy. The jump frequency can, therefore, be written as U ± = o exp − Em ∓ kB (3.78) 2

234


Em

u

S=0 S=1

Aβ + Bα = Aα + Bβ

Figure 3.60. Schematic diagram showing the energy barriers for the reactions of A and B atoms with vacancies on the and sublattices in a fully ordered (S = 1) and disordered (S = 0) alloy.

where kB denotes Boltzmann’s constant. The lower value of the activation energy holds for a vacancy jump in which an atom changes from a wrong to the right sublattice, i.e. for an ordering jump ("+). A higher value of activation energy refers to a disordering jump ("−). The overall rate of change of order in an irradiation environment can be written as (Banerjee and Urban 1984) a sum of three terms: dS/dt = dS/dtc + dS/dtr + dS/dtt

(3.79)

where (dS/dt)c and (dS/dt)r refer to the disordering rate due to the replacement collision sequence and the random defect annihilation process, respectively, while (dS/dt)t is the rate of change of order parameter arising from the thermally activated exchange of A and B atoms from the and sites and vice versa: dS/dtt = K+ cA cB − K− cA cB

(3.80)

where cA , c , etc. represent concentration of A in -site and B in -site, respectively, and the rate coefficients K+ (ordering) and K− (disordering) correspond to the reverse and forward reactions shown in Eq. 3.77 and are expressed as K± =

±Z cv Z cv Z cv + Z cv

where Z is the number of nearest neighbour -site around a -site.

(3.81)


235

0.010

0.3

T = 580 K

T = 1120 K 0.2

570

0.005

1130

dS/dt →

dS/dt →

0.1 0

1140

0

560

–0.1

553 540

–0.005 –0.2 –0.3

1150 0

0.1

0.2

Order parameter (S ) → (a)

0.3

–0.010

0

0.5

1.0

Order parameter (S ) → (b)

Figure 3.61. dS/dt versus S plots showing the influence of irradiation in creating a high concentration of anti-site defects for (a) thermal disordering and (b) irradiation disordering of a B2 alloy.

The influence of irradiation in creating a high concentration of anti-site defects (in other words, chemical disordering) can be illustrated by comparing the dS/dt versus S plots (Figure 3.61) for (a) thermal disordering and (b) irradiation disordering of a B2 alloy. While the disordering temperature, Tc , is 1140 K (above which dS/dt is negative for all values of S) under thermal disordering condition, the disordering temperature under irradiation Tc∗ is 553 K as shown in Figure 3.61(b). For details on kinetics of order–disorder transformation in alloys under irradiation, readers may refer to Banerjee and Urban (1984). The creation of anti-site defects (in other words, chemical disordering) plays a very important role in irradiation-induced amorphization, and the contribution of point defects is relatively less important. However, a quantitative estimation of the contributions of these components requires experiments using different types of radiations that have different replacement to displacement ratios and modelling of the irradiation-induced microstructural evolution using chemical rate equations as well as a molecular dynamics approach. Extensive experimental results on irradiation-induced amorphization of the Laves phase ZrCr Fe2 precipitates in a zircaloy-2 matrix under electron, ion and neutron irradiation are available. Some of the important results are summarized here with a view to making comparisons of the efficacy of different types of radiation with regard to bringing about amorphization: (1) Amorphisation occurs with all the three types of radiation when the irradiation (electrons, ions, neutrons) temperature is below a critical temperature, Tc . The radiation doses required for amorphization under 1.5 MeV electron, 127 MeV Ar ion and neutron irradiation are shown in Figure 3.62.

236


Electron

Dose (dpa)

60

Ion

Neutron

40

20

0

200

300

400

500

600

Temperature (K)

Figure 3.62. Irradiation dose (dpa) as a function of temperature (K) quantifying the radiation doses required for amorphization under 1.5 MeV electron, 127 MeV Ar ion and neutron irradiation.

Crystalline fraction Electron

1.00

0.50

Neutron 0.00

0.00

0.50

1.00

Fraction of irradiation time

Figure 3.63. Crystalline fraction as a function of irradiation time showing the sharp drop in the degree of crystallinity with electron irradiation.

(2) Under electron irradiation, amorphization occurs homogeneously within the entire volume of the precipitates. The degree of crystallinity drops sharply, as shown in Figure 3.63. Since electron irradiation is carried out on thin foils (10–20 nm thickness), interstitials migrate to the surface swiftly, allowing vacancy supersaturation to build up in the centre of the foil so that there is a substantial accumulation of defect energy within the foil. Chemical disordering also contributes towards amorphization.


237

(3) Under neutron irradiation, the amorphization of ZrCr Fe2 precipitates initiates from the precipitate–matrix interface, suggesting that cascade (or ballistic) mixing in the thin layer close to this interface is responsible for amorphization. The ballistic mixing of the two phases at the interface can bring about a significant departure from stoichiometry which causes a large increase in the free energy of this thin layer. With the increase in radiation dose, the amorphous layer gradually propagates towards the core of the precipitates, as shown in Figure 3.63.

3.8

PHASE STABILITY IN THIN FILM MULTILAYERS

When thin films are deposited on suitable substrate surfaces, they can exhibit crystal structures which are metastable with respect to those associated with the same materials in bulk form. Such metastable structures have been documented in literature for many systems including metal/metal and metal/semiconductor systems. Experimental observations on multilayers include instances where either one or both layers can exist in the metastable state. With the increase in introduction of multilayered nanostructures in a variety of applications, it is pertinent to examine the reasons for the shift in the relative stabilities of the relevant phases with variation in the thicknesses and in the thickness ratio of the constituents. Once again it is seen that the knowledge gained from recent researches on Ti- and Zr-based multilayer systems has provided an insight into this important issue. To illustrate this point, experimental observations on Ti/Al multilayers reported by Fraser and coworkers (Ahuja and Fraser 1994a,b, Banerjee et al. 1996) are summarized here. An examination of multilayered structures comprising alternate Al and Ti layers, each of several nm thickness, has revealed that in systems in which individual layers are of equal thickness and the unit bilayer thickness ≥ 20 nm, both metals assume their stable structures, namely fcc for Al and hcp for Ti. However, for < x nm, both metals have the hcp structure and for intermediate thickness ranges (x < y < ) both show the fcc structure. Thus, with increase in , the Al layer transforms once from hcp to fcc, while the Ti layer transforms twice: from hcp to fcc and then back to hcp. This behaviour of Ti is unexpected in two ways: first because it transforms from its stable structure (hcp) to a metastable structure (fcc) upon increase in thickness, and second, the transition from fcc/fcc to hcp/fcc multilayers occurs upon increase in thickness with a decrease in the misfit at the interfaces. The observations showing structural changes in multilayers have been rationalized by Banerjee et al. (1999) in the manner outlined here. A bilayer “unit system” comprising two layers and two interfaces (Figure 3.64) is defined and the specific free energy of this “unit system” is expressed as

238


A B A B A

Figure 3.64. A schematic diagram showing a bilayer “unit system” comprising two layers and two interfaces.

g = 2 + GA fA + GB fB

(3.82)

where g is normalized by the area of the interface, is the change in interfacial energy, Gi = G (metastable) − G (stable)) and fi are, respectively, the allotropic free energy change per unit volume of the reference phase for and the volume fraction of the metal i in the reference bilayer. Equation (3.81) describes a thermodynamic potential surface that varies as a function of two independent variables, f and . The specific free energy of this biphase system, as described in Figure 3.65(a), can be represented by a surface in the f − −1 space, and the equilibrium structure will be the one with the lowest specific energy of formation, g. The transformation from one biphase configuration to another (as e.g. Tifcc + Alfcc → Tihcp + Alfcc ) occurs when the g surface for a given combination of biphase (e.g. Tifcc + Alfcc ) intersects the g surface for a different biphase (e.g. Tihcp + Alfcc ). In this manner, the stability regimes of different biphase configurations can be depicted in biphase diagrams. It may be noted here that the interfacial energy term, , includes a chemical term arising out of dissimilar metals bonding at interfaces as well as a structural term due to the disregistry of the two contiguous lattices at the interfaces. The strain energy and the bulk chemical free energy terms are included in Gi which is volume dependent. Let us consider the case of the Al/Ti multilayers studied by Ahuja and Fraser (1994a,b) who reported a transition of Ti from the bulk, stable hcp form to a metastable fcc form below a critical value of = ∗fcc/fcc and both Al and Ti becoming hcp below another value of = ∗hcp/hcp . These two transformations of the biphase (Al/Ti) system can be depicted conveniently by a constant volume fraction cut of the g f surface, i.e. a plot of g versus , as shown in Figure 3.65(b). For an arbitrarily large value of , the lowest free energy of


239 fcc/fcc ΔGAl/Al

fcc/fcc

0.25

hcp/hcp fcc/hcp

(2)

0.2

(ii) (iii)

fcc/fcc

0.1

2Δγ hcp/hcp

hcp/hcp

0.05 0.0 0.0

hcp/fcc

2Δγ fcc/fcc

0.15

(1)

0.2

fcc/hcp

0.4

hcp Ti + fcc Al fcc Ti + fcc Al

hcp Ti + fcc Al

(3)

0.6

ΔGTi / Ti

(i)

0.0 Δg

1/λ,λ = bilayer thickness (nm)

hcp/hcp

0.3

0.8

λ∗hcp/hcp

1

λ∗fcc/fcc

Volume fraction of Ti2 fTi

Bilayer thickness, λ

(a)

(b)

a

B A

Ti

B A

Al

1 nm

Ti (c)

Figure 3.65. (a) The specific free energy of a biphase system represented by a surface in the f − −1 space; (b) g versus plot showing a transition of Ti from the bulk, stable hcp form to a metastable fcc form below a critical value of = ∗fcc/fcc and both Al and Ti becoming hcp below another value of = ∗hcp/hcp ; and (c) HRTEM image showing the Al/Ti multilayered sample, with the thickness of the Ti and Al layers being 5.0 and 20 nm, respectively. The beam direction is parallel ¯ to (after R. Banerjee et al.).

the system is achieved if both metals assume their stable structures, i.e. GTi =

GAl = 0 (line (i) in Figure 3.65(b)). Below = ∗fcc/fcc , the lowest free energy of the system corresponds to a bilayer configuration in which Ti has adopted a metastable (fcc) structure (line (ii) in Figure 3.65(b)). Below a still lower value of = ∗hcp/hcp , the energy of the system is minimized by the transition of both

240


metals to the hcp structure (line (iii) in Figure 3.65(b)). The stability diagram shown in Figure 3.65(b) provides a physical basis for the formation of metastable phases in nanolayered materials. In the hcp Ti + hcp Al regime, Ti remains in its stable state, making GTi = 0. For this regime, Eq. 3.82 is then reduced to

g = 2 hcp/hcp + GAl fAl

(3.83)

This means the intercept and the slope of the line (iii) are 2hcp/hcp and GAl fAl , respectively. Based on a similar argument, the intercept and the slope of the line (ii) are 2fcc/fcc and GTi fTi . The line (i) corresponds to the combination of hcp Ti and fcc Al, both in their respective stable states, implying that , GAl and

GTi are all equal to zero. It is, therefore, evident that the metastable states such as fcc Ti and hcp Al are stabilized in nanolayered structures due to the negative values of which more than compensate for the increase in G resulting from the formation of the metastable phases. Figure 3.65(b), which is consistent with experimental observations on the hierarchy of biphase stability, originates from the following inequalities:

GAl < GTi

and

hcpTi/hcpAl < fccTi/fccAl < 0

(3.84)

Figure 3.65(b) is constructed for a fixed value of the volume fraction, fTi . Biphase diagrams for cases where both the volume fraction and the bilayer thickness are variable can be constructed in the fi − −1 space. This is illustrated for Ti/Al multilayers in Figure 3.65(a). In the absence of coherency, the boundaries in this type of biphase diagrams are straight lines. Non-linear dependence of composite moduli on volume fraction will introduce curvature in the variation of 1/, with f in the biphase diagram. The biphase diagram for the Al/Ti multilayers has been constructed on the basis of the slopes of the lines (1), (2) and (3) respectively, which are given by GTi /2 fcc/fcc , GAl /2 hcp/hcp and ( GAl + GTi /2 hcp/hcp − fcc/fcc ). Experimental data points are superimposed on the biphase diagram of the Al/Ti multilayered system shown here. The symbols shown in the inset represent fcc Al/fcc Ti, hcp Al/hcp Ti and fcc Al/hcp Ti multilayers observed for different values of fTi and . These experimental data points are consistent with the biphase diagram. Application of the biphase diagram concept in predicting the structure of several multilayer systems such as Co/Cr, Zr/Nb and Ti/Nb has been successful (Thompson et al. 2003). A HRTEM image (Figure 3.65(c)) shows the Al/Ti multilayered sample with the thickness of the Ti and Al layers being 5.0 and 2.0 nm, respectively (Banerjee et al. 1996).


3.9

241

QUASICRYSTALLINE STRUCTURES AND RELATED RATIONAL APPROXIMANTS

Quasicrystalline structures can be defined as those with long-range aperiodic order and crystallographically forbidden rotational symmetries (e.g. 5-, 8-, 10- and 12fold rotation axes). The observation that certain intermetallic compounds exhibit sharp diffraction peaks displaying the “non-crystallographic” icosahedral rotational point group has generated a great deal of excitement. It is well known that the translational periodicity of atoms allows only certain rotational operations about an axis which bring the arrangement back into registry with the unrotated assembly. For three-dimensional periodic crystals, the allowed rotation operations are two-, three-, four- and sixfold, about appropriately chosen axes. Taken together with other operations such as translations, reflections and inversions, these point group operations define all of the 230 space groups. It was generally believed that only periodic arrangements of atoms can produce sharp diffraction peaks. The discovery of a quasicrystalline structure in a rapidly quenched Al–Mn alloy, schematic diffraction patterns from which are shown in Figure 3.66(a), has laid this myth (Shechtman et al. 1984) to rest. It is now realized that the occurrence of sharp Bragg diffraction peaks does not require the presence of long-range periodic translational order, but rather of long-range positional order, which may or many not be specified by a periodic function in three dimension. This point can be explained in one dimension by considering the Fibonacci sequence: 1 1 2 3 5 8 13 21 34# # # where every term of the series is generated by the addition of the two immediately preceding terms. Let us consider two translation vectors S (short) and L (long) along a given direction and generate a series using an algorithm in which S is replaced by L and L is replaced by LS in every successive series. It may be noted that the repeat period grows as per the Fibonacci sequence of numbers (as shown in Figure 3.67). The ratio of the number of L to the number of S segments√changes in recursive steps and finally converges to the “golden mean” ( = 1 + 5/2 1618) as the repeat period grows to infinity. This illustrates how an array of two segments, S and L, can be created in such a way that translational symmetry is absent even when the one-dimensional sequence is extended upto infinity. Penrose (1974) has shown that by using two specially shaped tiles, as designated by a kite and a dart (Figure 3.68), it is possible to cover a plane with fivefold symmetry. There exist “matching rules” for the construction of the Penrose lattice. The ratio of the sides of these tiles (kite and dart) is given by , as shown in Fig. 3.68. Some important properties of the Penrose pattern are (a) orientational order, (b) quasiperiodic translational order and (c) self-similarity.

242

0τ21 10.81°

°

79.2

13.28°

29°

τ2τ41

58.

°

(6)

.37

37

20.91°

31.72

(5)

63.43

13.28° (4)

20.91° 1τ20 (1)

(2)

010

10.81° 1 τ 0 (3)

(a)

(b)

Figure 3.66 Schematic diffractive pattern from a quasicrystalline phase showing (a) fivefold, threefold and twofold symmetries and the observed angles between the corresponding zone axes and (b) stereogram showing matching of observed symmetry elements with those of an icosahedron.


13τ + 1τ

Solidification, Vitrification and Crystallization A • • PERIOD: 1 • • A PERIOD: 2

A

B •

A

A • A/B = 1/0 = 00

•

A

•

•B •

A

• B • A/B = 1/1 = 1

(b)

• • A A/B = 2/1 = 2

(c)

(d)

A

•

243 (a)

A • B • • PERIOD: 3

A

•

A

•

B

A • B • • PERIOD: 5

A

•

A

•

B

A • B • • A/B = 3/2 = 1.5

B A • • • PERIOD: 8

A

•

A

•

B

B • A • • A/B = 5/3 = 1.66

H

T

A

• (e)

Figure 3.67. Fibonacci sequence of numbers.

T 1 H 72°

144°

1 72°

1 36°

H

1 36°

T

216°

τ

τ 72°

τ

τ 72°

T

H

(a)

(b)

Figure 3.68. Schematic diagrams showing two specially shaped tiles (a) kite and (b) dart. The ratio, , of the sides of these tiles (kite and dart) is also shown in the figure.

Mackay (1982) has extended Penrose tiling to three dimensions (3D) and has demonstrated that by making use of a pair of acute and obtuse rhombohedral tiles (as shown in Figure 3.69), filling of space with fivefold symmetry is possible. The interesting feature of 2D and 3D Penrose lattices is that the Fourier transform of their structures gives rise to sharp diffraction peaks displaying icosahedral symmetry for the latter. Quasilattices can be constructed with any arbitrary orientational symmetry and arbitrary quasiperiodicity. The existence of octagonal (Wang et al. 1987), decagonal (Bendersky 1985, Chattopadhyay et al. 1985) and duodecagonal (Ishimasa et al. 1985) quasicrystalline structures in various alloy systems has been established by experiments.

244


(b) (a) (c)

(d)

(e)

Figure 3.69. Penrose tiling in three dimension (3D) demonstrating filling of space with fivefold symmetry by using a pair of acute and obtuse rhombohedral tiles.

The diffraction patterns shown in Figure 3.66(a) show fivefold, threefold and twofold symmetries and the observed angles between the corresponding zone axes. The observed symmetry elements matched with those of an icosahedron as illustrated in the stereogram shown in (b). An inspection of the sequence of diffraction spots along a radial direction of the fivefold pattern in Figure 3.66 shows that the ratio of the distances from the origin to any two bright spots is an irrational number within reasonable experimental error. For icosahedral √ quasicrystals, this irrational number is some power of the golden mean, = 1+ 5/2, which arises


245

from the geometries of icosahedra, pentagons and decagons. Though translational symmetry is not present in these patterns, there exists an inflation symmetry. For example, the diffraction patterns in Figure 3.66 can be expanded or contracted by a factor of 3 to yield patterns indistinguishable from the originals. In view of the observed icosahedral symmetry of the diffraction patterns from some quasicrystalline structures, the indexing of these patterns has been carried out on the basis of six real space vectors defined by vectors, ei , that point from the centre to the vertices of an icosahedron (as illustrated in Figure 3.70). The reciprocal lattice vectors are then defined by G1 = 6i=1 ni ei ⎡ ⎤ 1 0 ⎢ −1 ⎡ ⎤ 0 ⎥ ⎢ ⎥ i ⎢ ⎥ 1 0 1 ⎥ ⎣ ⎦ ⎢ j ei = √ (3.85) 0 1 ⎥ 1 + 2 ⎢ ⎢ ⎥ k ⎣ 0 −1 ⎦ 0 1 − where ei are the real space basis vectors, ni are integers and go is a constant which determines the scale of the diffraction pattern. Unlike in the case of periodic crystals where the diffraction pattern can be related to the lattice parameter, a, by a relation of the type 2/a, no single fundamental length, go , can be chosen for diffraction patterns from quasicrystals ab initio. This is also evident from the presence of the inflation symmetry. The reciprocal space of the icosahedral quasicrystal, instead of having a regular lattice of intensity maxima as observed for crystals, can have peaks arbitrarily close to any given peak by taking integer linear combinations of fundamental basis vectors. Thus the reciprocal space of the icosahedral quasicrystal is uniformly 1

5

6

4

6

4

2

3

5

2 3 ei⊥

i eN

(a) 1 (b)

Figure 3.70. Schematic diagrams showing six real space vectors defined by ei that point from the centre to the vertices of an icosahedron.

246


dense. Within the Landau theory, one expects a decreasing hierarchy of peak intensities as the number of reciprocal lattice star vectors required to arrive at the peaks increases. Since the peaks close to any given peak are obtained only by the higher generation number, the intensities associated with these peaks are very feeble and are not distinguishable from the background. Peaks associated with lower indices are strong enough to produce a diffraction pattern with a discrete set of spots/reflections. For an icosahedral quasicrystalline structure, each reciprocal lattice vector requires six indices for indexing as has been expressed in Eq. 3.85. The advantage of describing a three-dimensional quasiperiodic structure using a six index system (which corresponds to a six-dimensional space) arises from the fact that a projection from a higher dimensional periodic lattice points on to a lower dimensional space can generate either a periodic or a quasiperiodic lattice, depending on the orientation of the projected space. This point can be explained in a simple manner by taking the example of a projection from a 2D space to a 1D space. Let us consider a 2D square lattice which is projected on a set of two perpendicular directions, designated as g1 and g2 . If the direction g1 is drawn from any lattice point of the 2D structure along a rational direction (defined by tan = m/n, where is the angle between g1 and the X-axis of the 2D lattice), the g1 line will intersect the lattice point with coordinates (n m) and will periodically intersect lattice points (2n 2m), (3n 3m), etc. resulting in a periodic one-dimensional structure. In contrast, if tan is an irrational number, the g1 , line starting from the origin will not intersect any other lattice point. For projecting lattice points of the 2D structure on to the line g1 , we can arbitrarily select a strip indicated by a pair of broken lines parallel to g1 as shown in Figure 3.71. Projections of lattice points lying within this strip on g1 produce an array of points which is quasiperiodic. Two segments, short and long, appear along g1 , but the sequence L S L L S L S L L S # # # is such that one cannot identify a unit which has a repeated periodic appearance. If the strip width is increased more number of spots will appear on the projected line g1 , making the line uniformly dense as the strip width is enlarged to infinity. However, it can also be seen that two points which are very close on the projected line will appear only if the strip width is increased to a very large extent, which means that these points arise from two points (in 2D lattice) which are widely separated along the g2 direction. A Fibonacci sequence in 1D can be created by projecting a 2D square lattice on a g1 line which has an inclination, tan = . In case tan is chosen to be a rational quantity, the projection will result in a periodic sequence. A particular set of crystalline approximants, the Fibonacci rational approximants, is obtained when the ratio of two consecutive numbers p and q of the Fibonacci sequence is


247

g⊥ g ⏐⏐

Figure 3.71. Schematic diagram indicating that projection of lattice points of 2D structure on to the line g1 . This can be done by arbitrarily selecting a strip marked by a pair of dashed lines parallel to g1 .

chosen (tan = q/p), where q/p = 1/1 2/1 3/2 5/3 8/5 138 21/13# # #

(3.86)

It is to be emphasized that the rational ratio is not restricted to the Fibonacci series since non-Fibonacci rational approximants have also been observed experimentally. The compositional similarities between quasicrystals and their respective approximants suggest similarities in their local atomic structures, substantiated by similarities in physical properties. Approximants are important for studies on the formation and stability of quasicrystals since they are amenable to established theoretical tools. Reversible transformations between quasicrystals and related crystalline approximant structures have been encountered in some cases in which the structural relationship between them could also be established. The projection method for the 1D case can be extended to 2D and 3D and can be used for the construction of both the real lattice and the reciprocal lattice. The 3D projected lattice represents the icosahedral quasilattice. Just as the reciprocal lattice of a crystalline structure is generated by a basis of three vectors, the icosahedral diffraction pattern is generated by a set of six reciprocal lattice vectors because of its incommensurate nature. This means that all the g vectors in the reciprocal space of an icosahedral structure can be expressed in terms of linear combinations of the basis vectors of the reciprocal space. In this method, a 6D periodic reciprocal lattice is projected orthogonally on to a suitably oriented 3D subspace.

248


For the icosahedral alloys, characterized by six fivefold axes, the 3D structure results from the projection of points of a 6D hypercubic lattice that are contained in a 3D acceptance domain (analogous to the strip drawn with a pair of broken lines for the 2D to 1D projection as shown in Figure 3.71) appropriately oriented with respect to the 6D lattice. This acceptance domain is a triacontahedron in the 3D space which is orthogonal to the physical space. For the icosahedral phase, the orientations of the acceptance domain and the physical space are specified by a 6 × 6 orientation matrix given by : 1 −1 0 0 1 0 0 1 1 0 1 −1 − 0 (3.87) M=√ 0 1 1 0 2 3 + 2 − 1 − 0 0 − 1 0 0 1 − −1 The upper three row vectors give the 6D coordinates of the three vectors which define the physical space while the lower three rows refer to vectors spanning the orthogonal space. The analogues of the L and S segments generated on the projection line from the points lying within the strip are oblate and prolate rhombohedra (Figure 3.69) which are the constituent tiles of the 3D Penrose lattice. The edge length of these rhombohedra, ar , plays a role analogous to the lattice constant for periodic crystals and is, therefore, called the quasilattice constant. As discussed in the case of the 2D to 1D projection, rational approximants can be constructed from the 6D hypercubic lattice by a suitable selection of the acceptance domain. Fibonacci rational approximants are obtained when is replaced by a rational ratio q/p. Elser and Henley (1985) first demonstrated that the cubic -AlMnSi structure can be obtained by a 1/1 rational projection from the same atomic decoration of the 6D hypercubic lattice used to define the icosahedral phases. They also made the first quantitative assessment of the similarities between the bcc (or close to the bcc structure) in the case of the -AlMnSi phase and the structure of the icosahedral phase. This analogy brings out the fact that the atomic arrangements of the related crystalline phases can be constructed from the same building blocks which generate the icosahedral phases. The building blocks may be taken either as icosahedral clusters of atoms or as a set of two types of rhombohedral bricks which may also be derived from the decomposition of the icosohedral clusters. 3.9.1 Icosahedral phases in Ti- and Zr-based systems Quasicrystalline structures have been reported most extensively in Al-based alloys. Icosahedral phase formation is now known to be quite common in Ti alloys


249

also, with Ti-based icosahedral phases constituting the second largest class of quasicrystals (Kim and Kelton 1995). Alloys containing Ti and 3d transition metals from V to Ni have received the maximum attention. Quasicrystalline phases have been reported in many of these alloys in the rapidly solidified condition. In most cases, the microstructure produced consists of a finely distributed mixture of quasicrystalline and crystalline phases. The presence of Si and O in these alloys often plays a crucial role in stabilizing the icosahedral phase. A representative list of alloys based on Ti and Zr in which the formation of icosahedral phases has been reported is given in Table 3.8. A number of alloys in which both Ti and Zr are present have attracted considerable interest (Kelton et al. 1994, Kim and Kelton 1995, 1996) due to their strong tendency for icosahedral phase (i-phase) formation. Some of these (such as Ti–Zr–Fe alloys) show localized diffuse scattering and significant diffraction spot shape anisotropy while some others (such as Ti–Zr–Ni alloys) are more ordered, as reflected in the sharp diffraction spots obtained from the icosahedral phases in these alloys.

Table 3.8. Icosahedral phases in Ti- and Zr-based alloys Alloy composition (approximate)

Processing/ stability

Extent of phason disorder

Quasilattice parameter (nm)

Ti-TM-Si-O

RSP/MS

Very high

0.47–0.48

Ti–Zr–Fe

RSP/MS

Very high

0.485–0.488

Ti53 Zr27 Ni20

RSP/MS

Very low

0.512

Ti53 Zr27 Co20

RSP/MS

Moderate

0.510

Ti415 Zr415 Ni17

AI/S

Very low

0.517

Ti63 Cu25 Al12

C/MS

Low

–

Zr65−70 Cu12−17 Ni10−11 Al75

C/MS

Low

–

Zr65−70 Cu10−15 Ni10−13 Pd7−10

C/MS

Low

–

Zr65 Cu125 Ni10 Al75 M5

C/MS

Low

–

M = Ag, Pd, Au, Pt RSP, rapid solidification processed; AI, annealed ingot; C, crystallized glass; S, Stable; MS, Metastable.

Reference Libbert and Kelton (1995) Kim and Kelton (1995) Zhang et al. (1994) Kim and Kelton (1996) Kim et al. (1997) Koster et al. (1996a) Koster et al. (1996b) Murty et al. (2000a) Murty et al. (2000b,c)

250


As can be seen from Table 3.8 the icosahedral phases in Ti-based systems can be grouped into two classes. Those belonging to the first are associated with a smaller quasilattice constant (∼0.48 nm) and show arcing of diffraction spots and diffuse intensity distribution, suggesting the presence of phason disorder to a considerable extent. In contrast, members of the second group have larger quasilattice constants (> 0.51 nm) and exhibit sharp diffraction spots indicating a higher degree of phason order. Out of the Ti-based icosahedral phases listed in Table 3.8, the i-phase in only Ti415 Zr415 Ni17 , which forms during annealing of arc-melted ingots, is stable. In all the other cases, the i-phase is metastable, forming during rapid solidification and disappearing during subsequent annealing. The microstructures of most of Ti-based icosahedral phases are similar. Fine particles of the i-phase are usually found dispersed in the amorphous matrix of rapidly solidified alloys of compositions listed in Table 3.8. These particles also appear during the early stages of crystallization of amorphous alloys. In a number of observations, i-phase particles have been found to be surrounded by the -Ti (bcc) phase. Several crystalline approximants of the i-phase are found to coexist in partially crystallized samples. The occurrence and the structure of these phases are briefly discussed in the following paragraphs. The -1/1 rational approximant, a large unit cell bcc phase (lattice parameter = 1.31 nm), consisting of Mackay icosahedra packed face to face along the cubic direction, is frequently observed in Ti-Mn-Si and Ti-Cr-Si alloys. This phase is believed to be the appropriate approximant to the Ti-3d TM-Si icosahedral phases. The -phase, a face-centred orthorhombic phase with a large unit cell (a = 320 nm, b = 266 nm, c = 104 nm) appears frequently with the i-phase in Ti-MnSi, Ti-Mn-Fe-Si and Ti-Cr-Si alloys. Based on TEM studies, it has been inferred that the -phase can be constructed structurewise from Mackay icosahedra packed with their vertices aligned along the c-direction of the unit cell. A large unit cell fcc phase (a = 1.12–1.17 nm), presumably with the Ti2 Ni structure, has been found in many Ti-Zr-Fe samples. Usually the fcc phase and the i-phase are found in different regions of the sample, suggesting that these two phases evolve directly from the liquid phase under different conditions of cooling. A hcp phase with a = 0.515 nm and c = 0.837 nm has been detected in TiZr-Fe alloys containing relatively low concentrations of Ti. Energy dispersive spectroscopy has revealed the composition of this phase to be Ti41−49 Zr21−29 Fe26−31 with small amounts of Si (Kim and Kelton 1995). This phase is isostructural with MgZn2 type Laves phases. The observed intensity modulation in the diffraction patterns from this Laves phase is similar to that associated with the i-phase, suggesting a similarity in the local atomic arrangements in the two.


251

As mentioned earlier, the i-phase is often seen to be surrounded by the (bcc)-phase. In addition to the prominent bcc reflections, spots at 1/3 and 2/3 positions corresponding to the fundamental bcc reflections are recorded. These additional spots suggest the presence of the -phase in the -matrix. The presence of the -phase is not unusual in -Ti alloys which contain sufficient amounts of -stabilizing elements. Unlike in the cases of Ti-TM-Si-O and Ti-Zr-Fe alloys, the i-phase forming in the Ti-Zr-Ni system is stable. This has been demonstrated conclusively by forming the i-phase by annealing as cast ingots of a Ti45 Zr38 Ni17 alloy which initially contained only a C14 Laves phase and a hexagonal solid solution phase. The crystalline approximant which forms along with the i-phase in Ti-Zr-Ni is the bcc W-phase (which is a 1/1 rational approximant). The composition range of the W-phase is Ti40−50 Zr31−42 Ni16−19 . XRD peak intensities from this phase are quite distinct from those pertaining to the -1/1 approximant described earlier. Based on the powder diffraction data and the number of atoms per unit cell (168.5, estimated from measured density), Kim et al. (1997) have inferred that the W-phase has a Bergman type structure similar to that encountered in Al-Li-Cu and Al-Mg-Zn alloys. In this context, it is worth mentioning that two basic cluster types, both having icosahedral symmetry, are used in describing atomic positions in icosahedral phases. The Mackay cluster, as shown in Figure 3.72 is a double-shell icosahedral cluster, with atoms decorating the vertices of the inner and outer icosahedra and the midpoints of the edges of the outer icosahedron (Mackay 1962). While the

-1/1 rational approximant in Ti-TM-Si-O alloys is based on the Mackay cluster, the W-phase in Ti-Zr-Ni alloys is based on the Bergman cluster which is also a double-shell icosahedral cluster; however, the midpoints of the faces of the outer Al Mn

Figure 3.72. The Mackay cluster – a double-shell icosahedral cluster with atoms decorating the vertices of the inner and outer icosahedra and the midpoints of the edges of the outer icosahedron.

252


icosahedron are occupied, instead of the edge centres as in the case of the Mackay cluster. The i-phases in these two types of systems can thus be grouped into two distinct classes, the former containing Mackay clusters and the latter constituted of Bergman clusters. Kim et al. (1997) have classified i-phases based on a correlation between the measured quasilattice constant, aq , and the atomic separation, as , calculated from the measured i-phase densities. By this method, all three Bergman type i-phases, including i-(Al-Li-Cu), i-(Al-Mg-Zn) and i-(Ti-Zr-Ni) have the ratio aq /as 2, while for i-phases which have Mackay type clusters, aq /as = 185.

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Chapter 4

Martensitic Transformations 4.1 Introduction 4.2 General Features of Martensitic Transformations 4.2.1 Thermodynamics 4.2.2 Crystallography 4.2.3 Kinetics 4.2.4 Summary 4.3 BCC to Orthohexagonal Martensitic Transformation In Alloys Based on Ti and Zr 4.3.1 Phase diagrams and Ms temperatures 4.3.2 Lattice correspondence 4.3.3 Crystallographic analysis 4.3.4 Stress-assisted and strain-induced martensitic transformation 4.4 Strengthening Due to Martensitic Transformation 4.4.1 Microscopic interactions 4.4.2 Macroscopic flow behaviour 4.5 Martensitic Transformation in Ti–Ni Shape Memory Alloys 4.5.1 Transformation sequences 4.5.2 Crystallography of the B2 → R transformation 4.5.3 Crystallography of the B2 → B19 transformation 4.5.4 Crystallography of the B2 → B19 transformation 4.5.5 Self-accommodating morphology of Ni–Ti martensite plates 4.5.6 Shape memory effect 4.5.7 Reversion stress in a shape memory alloy 4.5.8 Thermal arrest memory effect 4.6 Tetragonal Monoclinic Transformation in Zirconia 4.6.1 Transformation characteristics 4.6.2 Orientation relation and lattice correspondence 4.6.3 Habit plane 4.7 Transformation Toughening of Partially Stabilized Zirconia (PSZ) 4.7.1 Crystallography of tetragonal → monoclinic transformation in small particles References

260 261 261 266 277 280 281 282 289 294 324 326 329 335 339 340 342 342 345 347 352 356 360 362 362 363 366 369 372 373


Chapter 4

Martensitic Transformations

Symbols F: T: P: To : Ms : : as : f : a : : : Ep : E : Ms : i : i : Bi : R: S: P: E: x: x: L, B, T :

: kB : MB : Mf : R: Tan : : eijT : li :

and Abbreviations Helmoltz free energy Temperature Pressure Equilibrium transformation temperature Martensite start temperature Molar volume Surface area Chemical free energy change Applied stress tensor Microscopic strain Interfacial energy per unit area Total energy dissipated in plastic flow Elastic modulus of parent phase Stress required for martensitic transformation Principal strain with direction Principal distortions in the principal direction Bain strain matrix Rigid body rotation matrix Total shape strain matrix Lattice invariant shear matrix Total distortion matrix Composition Distance Plate dimensions (length, breadth and thickness) Twin fraction Boltzman constant Martensite burst temperature Martensite finish temperature Gas constant Magnitude of shear Poison ratio Stress free transformation strain Direction cosine of the position vector 259

260


V: Uj (s): DSA: IPS: LIS: b: Ms : Md : : : : p : Txy : h: : :

1 2 : stat : m : ∗: A∗ : T: As : Af : : T :

4.1

Volume of the inclusion Displacement at point x Degree of self-accommodation Invariant plane strain Lattice invariant shear Burgers vector Temperature above plastic yield starts after martensitic transformation Stress required for stress-assisted martensite nucleation Difference between the flow stresses of and Flow stress of Flow stress of True plastic strain Stress required to move a dislocation out of a small angle boundary Spacing between the dislocations Shear modulus Number of dislocations Geometric slip distances for and Independent component of flow stress Athermal component of flow stress Thermal component of flow stress Activation area Homogeneous deformation matrix Austenite start temperature Austenite finish temperature Reversion stress Reversion temperature

INTRODUCTION

Martensitic transformations take place in numerous materials. Evidences of their occurrence have been found in several pure metals such as Fe, Co, Hg, Li, Ti, Zr, U and Pu, in many ferrous and non-ferrous alloys and in several oxides and intermetallic compounds such as ZrO2 , BaTiO3 , V3 Si, Nb3 Sn, NiTi and NiAl. Some years ago, the word “martensite” was used solely to describe a microconstituent in quench hardened steels. Bain (1924) put forward a mechanism for the transformation of the face centred cubic austenite to the body centred tetragonal martensite


261

in steels in which the structural change was considered to be brought about by a homogenous deformation of the parent lattice. It was implicit in this description that the transformation did not involve random or diffusive atom movements and that it resulted from only small relative displacements of neighbouring atoms. The fact that a similar mechanism is operative in a large number of solid state phase transformations has led to a proliferation of the use of the terms “martensite” or “martensitic” in a much wider sphere. As indicated in Chapter 3, martensitic transformations are grouped in the general class of displacive transformations and belong to that subset which involves the operation of a lattice deformation. The characteristic features of martensitic transformations are described in the following section to provide a background for discussions on martensitic transformations in alloys, intermetallics and ceramics based on Ti and Zr.

4.2

GENERAL FEATURES OF MARTENSITIC TRANSFORMATIONS

Martensitic transformations are characterized by a number of thermodynamic, kinetic, crystallographic and mechanistic features. Experimental observables pertaining to each of these are needed to unequivocally qualify a transformation to be martensitic in nature. This section is devoted to a brief discussion on these aspects. 4.2.1 Thermodynamics The first and foremost condition of a martensitic transformation is that the product phase inherits the composition of the parent phase. For a single component system like a pure metal, the driving force for the transformation to occur can be represented with the help of Helmholtz free energy (F ) versus temperature (T ) or pressure (P) plots. Taking the example of pure Fe, the chemical free energy change, F , accompanying a transformation from the austenite to the ferrite phase, can be expressed as follows: F = −1202 + 263 × 10−3 T 2 − 154 × 10−6 T 3 cal/mol 200 K < T < 900 K F = −1474 + 340 × 10−3 T 2 − 200 × 10−6 T 3 cal/mol 800 K < T < 1000 K (4.1) While the former expression is due to Kaufman and Cohen (1956, 1958), the latter was proposed by Owen and Gilbert (1960). This change in free energy

262


ΔF α→γ = 1202–2.63 × 10–3T 2 + 1.54 × 10–6T 3 Fe

ΔF Fe α→γ (cal/mol)

1600 1400 1200 1000 800 600 400

0

100

200

300

400

500

600

700

800

Temperature (K)

Figure 4.1. The free energy change accompanying the → in pure Fe as a function of temperature.

keeps on increasing with a lowering of the temperature from the austenite/ferrite equilibrium transition temperature, 1183 K (Figure 4.1). When a multicomponent system is considered, the chemical driving force is given by the drop in the free energy of the system as the parent phase transforms into the product, retaining the initial chemical composition. In this sense, the system behaves as if the transformation is occurring in a single component system. This point can be illustrated by taking the example of the Fe–Ni alloy system, the phase diagram and the free energy – composition diagrams which are shown in Figure 4.2(a) and (b), respectively. The equilibrium condition between the austenite and the ferrite phases can be identified by constructing a common tangent which locates the compositions of the two phases in equilibrium. When such partitioning of the alloying element is suppressed by a rapid quench, the composition, xo , can be defined at which the integral molar free energies of the two phases are equal at To . A composition-invariant transition from austenite to ferrite is thermodynamically possible only below this To temperature. The composition dependence of To is superimposed on the phase diagram in Figure 4.2(a). Martensitic transformations, like any other first-order transformation, do not start at To , where F = 0 but are initiated when some supercooling is provided. The temperature at which a martensitic transformation “starts” is known as the Ms temperature. The difference, To − Ms , indicates the extent of supercooling required

+1200

x=0 x = 0.05

+1000

–1000

x = 0.10 x = 0.15

+800

–800

Ms

x = 0.35

+200

–400 –200

0

0

–200

+200

–400

+400

–600

+600 400

600 800 1000 1200

Temperature (K) (a)

Ms ½ (Ms + As) γ

800 700 600

α+γ

Ad

500 400

α

α+γ

300

As

200

As

1100

900

–600

ΔF γ→α′ = F α′– Fγ Temperature (K)

ΔF α→γ = F γ – Fα′

+400

x = 0.25 x = 0.30

1200

1000

x = 0.20

+600

263

–1200

cal/mol

cal/mol


Md

To (calc.)

200 100 00

Fe

10

20

30

40

50

x (Ni at. %) (b)

Figure 4.2. (a) Experimental and theoretical determination of To in the Fe–Ni system. (b) Chemical free energy change accompanying the martensite transformation in the Fe–Ni system.

to initiate the transformation. For a number of ferrous alloys, this difference is about 200 K, while for the alloys based on Ti and Zr, it is much lower (≈50 K). The requirement of supercooling arises from the necessity of overcoming the following energy components which oppose the transformation: (a) the interfacial energy between the martensite and the parent matrix, (b) the elastic energy stored in the martensite – parent assembly to accommodate the shape change and the volume change accompanying the transition, (c) the energy dissipated in plastic deformation of both the martensite and the parent phases and (d) the driving force required for the rapid propagation of the martensite interface. Apart from the chemical free energy change, F , an applied stress can contribute to the driving force for a martensitic transformation. This is amply demonstrated in several alloy systems where the transformation can be induced at a temperature higher than the Ms temperature by applying stress. This effect can be attributed to the interaction of the applied stress field with the shape strain involved in the formation of a martensite plate. Provided the interaction has the correct sign, the formation of a plate will relieve the potential of the applied stress field. The formation of a small region of martensite in the presence of a stress field will release a small amount of mechanical energy, which may be positive or negative depending on the nature of the stress field and the orientation of the plate.

264


Considering the driving forces arising due to the chemical free energy change and the applied stress and the restraining forces associated with the four factors listed earlier, the following energy balance expression can be written for the initiation of a martensitic transformation: vf + A1 ≥ As + vA2 E 2 + Ep

(4.2)

where the volume and the surface area of the martensite plate are denoted by v and as , respectively, f is the free energy change per unit volume of the martensite, a and are tensors representing, respectively, the applied stress and the macroscopic strain associated with the transformation, is the interfacial energy per unit surface area of the plate, E is the elastic modulus of the parent phase, Ep is the total energy dissipated in plastic flow and in imparting a high velocity to the martensite interface and A1 and A2 are dimensionless geometrical factors. The specific interfacial energy, , between the martensite and the matrix depends on the extent of coherency at the interface. Since atom transfer across the transformation front (interface) occurs through coordinated and highly disciplined atom movements (like regimented movements during a change in a military formation), the maintenance of coherency at the interface becomes a necessary condition. The presence of an array of dislocations at the interfaces arises out of a geometrical necessity as will be discussed in a later section. The changes in the shape and in the specific volume associated with the formation of a martensite plate of a given geometry in the matrix result in the development of a strain, both within the plate and in the matrix. The partitioning of the strain between the two phases, however, depends on the respective values of their elastic moduli. The strain so developed is accommodated either by an elastic deformation of the assembly or by a combination of plastic flow and elastic deformation. The latter situation prevails when the accommodation stress developed exceeds the flow stress in either of the phases. The driving force for the martensitic transformation must exceed the corresponding restraining force for the growth of the transformation product. The difference between the driving and the restraining forces is utilized in moving the interfacial dislocations. For a conservative movement of dislocations, the Peierls stress and the other internal stresses opposing their motion need to be overcome. The growth velocity of martensite interfaces, measured by the rate of change of electrical resistance of samples undergoing a martensitic transformation, has been found to be about one-third the velocity of elastic waves in the parent phase. This growth velocity was also found by Bunshah and Mehl (1953) to be essentially constant at all temperatures between 73 and 293 K for both types of martensites which show athermal and isothermal kinetics for overall growth. The facts that the growth


265

velocity is very high even at cryogenic temperatures and that it is independent of temperature suggest that the growth process is athermal. Rapid growth is commonly encountered when the transformation is driven by large driving forces and is thus adiabatic. The interface can, therefore, accelerate rapidly up to its limiting velocity, which is of the same order as the velocity of crack propagation or of twin formation. Such a growth of an isolated plate can cause plastic deformation in the matrix which, in turn, results in the loss of coherency at the interface and in the nucleation of fresh plates in the adjoining untransformed regions. The growth of the primary plate ceases at this point. Once the coherency at the interface is lost, it is not possible to reactivate its motion by changing the driving force (either by heating/cooling or by deformation). In contrast to the scenario described above, a martensite plate can reach a thermoelastic equilibrium when it assumes its full size under a given condition of temperature and applied stress. This can happen if the driving force (having chemical as well as mechanical components) exactly balances the restraining force arising from the surface energy and the elastic strain energy. The basic requirements for attaining a thermoelastic equilibrium are, therefore, that the elastic stress limits in the parent and the product phases should be high and that the shape strain associated with the transformation should be small. As the strain energy builds up with the growth of a plate, a thermoelastic equilibrium is established when the plate assumes a certain critical size. In such a situation, the interface retains complete coherency and is amenable to movement in either direction, leading to the growth or the shrinkage of the plate, depending on the magnitude of the driving force. As pointed out earlier, a supercooling to the extent of To −Ms is needed to induce spontaneous nucleation of martensite plates. Martensite plates can, however, be nucleated at temperatures higher than Ms if additional driving force is provided by an applied stress. The influence of such an applied stress on the martensitic transformation can be explained by using a schematic diagram (Figure 4.3) which was originally presented by Olson and Cohen (1972). It can be seen that the stress required for martensite formation increases linearly as the temperature rises from Ms to Ms ; beyond this point, plastic deformation of the parent phase sets in. In the temperature range, Ms < T < Ms , the applied stress complements the chemical driving force which decreases linearly with increasing temperature. Martensite nucleation in this temperature range is stress-assisted. At temperatures higher than Ms , the elastic driving force derived from the applied stress is inadequate to satisfy the requirement for martensitic nucleation. As the applied stress exceeds the flow stress of the parent phase, plastic deformation causes the creation of fresh martensite nuclei and the formation of strain-induced plates. The features shown in Figure 4.3 have been explained in detail in Section 4.3.4.

266


b

Str tra ain-i n s nd for uc ma ed tio n

σ2

a

Str tra ess-a ns for ssis ma ted tio n

Applied stress

c

σ1

Ms

T1

0.2% pro of au of stress stenit e

T2 Msσ Temperature

Md

Figure 4.3. Schematic diagram showing the critical stress for martensite formation in a typical ferrous alloy as a function of temperature.

4.2.2 Crystallography The crystal geometry associated with martensitic transformations in various systems has been found to be governed by invariant plane strain (IPS) considerations which will be discussed in this section. The validity of the IPS criterion in predicting the transformation geometry is so overwhelming that sometimes a transformation is identified as martensitic purely on the basis of this geometrical criterion. This approach, however, is currently being questioned since some diffusional transformations have also been shown to exhibit geometrical features predictable from IPS considerations. In this section, the essential points concerning the phenomenological theory of martensite crystallography, developed independently by Wechsler et al. (W-L-R)(1953) and by Bowles and Mackenzie (B-M) (1954), will be discussed. The important geometrical features of martensitic transformations are listed below: (1) The formation of a martensite plate in a grain of the parent phase creates upheavals (surface relief) on a polished reference surface of the parent grain. This is illustrated in a schematic drawing in Figure 4.4 which shows the macroscopic shear produced in a parent crystal in which a martensite plate is formed. Observations on the displacement of reference lines drawn on the surface of the crystal indicate that all reference straight lines are transformed


267

D M4

C P

M3

P4

M2 P3

A

e nit

Ma r

P1

B

ste Au M ten sit e

M1

P2

P

ite

ten

s Au

Figure 4.4. The shape deformation due to formation of a martensite plate. Surface M1 M2 M3 M4 remains plane and tilted about M1 M2 and M3 M4 . The straight line AD marked on austenite is transformed into ABCD, where the segment BC within the martensite plate remains a straight line after the transformation. There is no discontinuity at points B and C, which are at the martensite– austenite interface, indicating that the interface is undistorted and unrotated.

into straight lines and all reference planes into planes in the product martensite. This implies that the transformation strain is linear and, therefore, can be expressed in the form of a matrix. Such a transformation is described mathematically as an affine transformation. The fact that no discontinuity is produced at the interface plane separating the martensite plate and the matrix indicates that the interface plane (habit plane) is an undistorted and unrotated plane (invariant plane). (2) The habit plane which is seen to be characteristic of a specific transformation is generally irrational. (3) A precise reproducible orientation relation is invariably present between the parent and the martensite crystals, as revealed from diffraction experiments. (4) Martensite plates very often contain a periodic arrangement of internal twins. The concept of lattice strain which came from the suggestion of Bain (1924) is illustrated schematically in Figure 4.5 wherein the fcc austenite is converted to the body centred cubic (bcc) ferrite by a single “upsetting” process in which the dimensions of the fcc unit cell are altered to those of the bcc unit cell by a homogeneous deformation of the parent lattice requiring only small shifts in the atom positions. The three vectors chosen to define unit cells in this description are mutually perpendicular before and after the lattice transformation. In general,

268

Phase Transformations: Titanium and Zirconium Alloys [101]A → [111]M

X3, X′3

(101)A → (112)M

X′2

X1

X2

X′1 (a)

ao

c

a o / √2 (b)

a (c)

Figure 4.5. Lattice correspondence and lattice deformation for the fcc to bct austenite–martensite transformation in Fe alloys.

in a homogeneous deformation, it is always possible to select three mutually perpendicular vectors (say, X1 , X2 and X3 ) which remain perpendicular after the deformation, and these are called the principal axes of deformation. When a volume of the parent phase, represented by a unit sphere, is subjected to a homogeneous strain, it is transformed into an ellipsoid. The construction of strain ellipsoids (Figure 4.6) illustrates the conditions for the homogeneous strain to have at least one plane undistorted. When the principal strains associated with the homogeneous strain are all positive or all negative, the strain ellipsoid does not intersect the unit sphere at all, implying that not a single vector remains undistorted by the homogeneous strain. Such a situation is shown in Figure 4.6(a). For a plane to remain undistorted, the necessary and sufficient conditions are that one of the principal strains should be zero while the other two should be, respectively, positive and negative. If the principal strains, 1 , 2 and 3 , are such that 1 is zero,

Martensitic Transformations X3

269 X3

B′

η3 = l+ε3

A′ A

B

l

η2 = l+ε2

(a)

X2

X2

(b)

Figure 4.6. Deformation of a unit sphere into an ellipsoid by homogeneous lattice strain (Bain strain) (a) 1 2 3 > 1 and (b) 1 = 1, 2 < 1, 3 > 1. The details are explained in the text.

2 is negative and 3 is positive (as illustrated in Figure 4.6(b)), the strain ellipsoid will touch the sphere at the point of intersection of the X1 axis with the unit sphere and will intersect the sphere at two points A and B on the plane containing the X2 and X3 axes. The planes defined by OA × X1 and OB × X1 vectors remain undistorted though they are rotated from their original positions, defined by the planes OA × X1 and OB × X1 vectors. An examination of the Bain strain necessary for the deformation of the parent lattice into the product lattice reveals that, in general, the Bain strain or lattice strain alone does not satisfy the aforementioned conditions which ensure at least one undistorted plane. Moreover, the macroscopic shape strains measured from the surface relief observations in several martensites do not match with the respective Bain strains. It is because of these two factors that the concept of a second shear was invoked in the martensite crystallography. While the Bain strain is responsible for bringing about the change in the lattice, the second shear, observed as the lattice invariant shear (LIS), superimposed on the Bain strain makes the total shear satisfy the undistorted plane condition. The necessity of a second shear can be explained by citing a specific example. The lattice or Bain distortion, B1 , necessary for transforming the (fcc) austenite

270


(with lattice parameter ao ) into the (bct) martensite (with lattice parameters a and c) can be expressed in terms of a matrix: ⎡

1 B1 = ⎣ 0 0

0 2 0

⎤ 0 0⎦ 3

(4.3)

√ where 1 = 2 = a 2/ao and 3 = c/ao Substituting the lattice parameter values for a carbon steel, one finds that a tensile strain of 12% in all directions perpendicular to the c-axis (X3 -axis, marked in Figure 4.5) and a compression of 17% along the c-axis are required for upsetting the lattice from fcc to bct. It is obvious that this lattice deformation cannot satisfy the condition for having an undistorted plane. Therefore, the total macroscopic shear, which is experimentally shown to be an IPS, must consist of additional components which, in conjunction with the lattice shear, satisfy the IPS condition. The same conclusion was arrived at, before the phenomenological crystallographic theory (W-L-R and B-M) was introduced, through an elegant experiment by Greninger and Troiano (1949). They experimentally determined the magnitude of the macroscopic shear from observations on the surface relief produced due to the martensitic transformation in an Fe–22% Ni–0.8% C alloy. They noticed that the experimentally measured macroscopic shear, when applied to the parent austenite lattice, did not generate the martensite lattice. In order to account for the observed difference between the macroscopic strain and the lattice strain, an LIS (either slip or twinning) has been introduced as a component of the total strain. The phenomenological theory of martensite crystallography is based on the postulate that the habit plane (the interface separating the parent and the product phases) is not an atomistically flat plane which remains invariant on a microscopic scale during the transformation. Misfits between the two structures develop and the accumulated misfits periodically get corrected to establish an average or macroscopic fit. The essence of the theory can be described by a set of schematic drawings (Figure 4.7). Let us consider the transformation of the two-dimensional lattice shown in Figure 4.7(a) to that shown in Figure 4.7(b), the corresponding unit cells being indicated by thick lines. The required lattice strain which brings about the change in the lattice also produces a shape strain; this is reflected in the rotation of the vector AB in the parent lattice to the vector A B in the product lattice. The magnitude of the vector A B can be brought back to the magnitude of the vector AB without changing the product lattice by the introduction of an LIS either by slip or by twinning. The geometries associated with these options are illustrated in Figure 4.7(c) and (d), respectively. In the case of the LIS being provided by slip, the product martensite plate consists of a single


271

B′

B

A Initial crystal (a)

A′

After lattice deformation (b)

B′

B′

A′ Lattice deformation followed by slip shear (c)

A′ Two lattice deformations leading to twin related regions (d)

Figure 4.7. Schematics showing (a) untransformed crystal, (b) after undergoing a lattice deformation, (c) the additional effect of a slip shear and (d) crystal having alternately twined regions. (c) and (d) show that a combination of lattice deformation and lattice invariant deformations (slip or twin) can make the habit plane an invariant plane.

variant of the martensite crystal. However, if the LIS is provided by twinning, two twin-related martensite variants form within a single martensite plate. In order to bring the vector A B into coincidence with the original vector AB, an additional rigid body rotation is necessary. The total macroscopic shape strain (S), which has to satisfy the IPS criterion, is, therefore, conceptually divided into components, namely the lattice strain (B) which is responsible for changing the parent lattice into the product lattice, the LIS (P), which, on being superimposed on the lattice shear, establishes an undistorted plane, and a rigid body rotation (R), which ensures that the undistorted plane is unrotated as well, S = RPB. In the case of twinning as the LIS, it is necessary to satisfy another symmetry criterion. The two twin-related orientations in the product phase evolve from a single parent phase crystal. It is, therefore, necessary that crystallographically equivalent lattice strains are operative in the adjacent regions which transform into a pair of twin-related product orientations of the product crystals. The formation of such a configuration is also expected from the consideration of symmetry breaking in a phase transformation. If the number of symmetry elements of the parent crystal gets reduced due to a transformation process, there is a general tendency for the restoration

272

Phase Transformations: Titanium and Zirconium Alloys B2

Mirror plane

φ2

B2 B D1

C2 C1

φ1 2

C2 C

B1

C1 A2

A1

1

A2 A1

D2 D B1

Figure 4.8. Schematics showing restoration of the symmetry in a macroscopic sense through the creation of a number of crystallographic variants.

of the symmetry in a macroscopic sense through the creation of a number of crystallographic variants. This can be illustrated in a two-dimensional construction (Figure 4.8) in which a parent square lattice ABCD is transformed into two equivalent rectangular lattices, A1 B1 C1 D1 and A2 B2 C2 D2 . When these two rectangular regions are rotated to bring them into coincidence along their diagonals, A1 C1 and A2 C2 , a twin is created where the twin plane is derived from a mirror plane in the parent crystal. In fact, the mirror symmetry of the parent crystal on this plane is lost due to the transformation, and the formation of the twinned product crystals tends to restore, at least partially, the lost mirror symmetry. The relative volumes of the two orientations, usually expressed in terms of the ratio of the thicknesses of the adjacent twins, are determined by the requirement of the lattice invariant deformation necessary to satisfy the IPS condition. Referring back to the transformation described in Figure 4.5, the lattice (Bain) distortions, B1 , associated with the two adjacent twin-related variants can be represented in the (i1 j1 k1 ) and (i2 j2 k2 ) principal axes systems respectively by the matrices ⎡ ⎡ ⎤ ⎤ 0 0 1 0 1 0 B1 = ⎣ 0 2 0 ⎦ and B2 = ⎣ 0 3 0 ⎦ (4.4) 0 0 3 0 0 2 These two matrices can then be expressed in the axis system of the parent crystal, (i j k), by the standard similarity transformation procedure, which involves


273

rotations of the axes systems from the basis of the martensite crystal to that of the parent crystal. The Bain strains, B1 and B2 , can be represented in the axis system (i j k) as B1 and B2 , respectively: ⎡ ⎤ ⎡ + ⎤ 2 − 1 1 + 2 2 − 1 1 2 0 0 ⎢ 2 2 ⎥ ⎢ 2 ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎥ and B2 = ⎢ 1 + 2 2 − 1 0 3 0 ⎥ B1 = ⎢ ⎢ ⎥ ⎢ 0⎥ ⎢ ⎥ ⎣ 2 ⎦ 2 ⎣ − 1 + 2 ⎦ 2 1 0 0 0 3 2 2 (4.5) In order to bring the two adjacent regions into twin-related orientations, it is necessary to introduce rigid body rotations, 1 and 2 , to the regions marked 1 and 2, respectively, in Figure 4.8. Therefore, 1 and 2 describe the rotations of the principal axes of the pure distortions in regions 1 and 2 relative to an axis system fixed in the untransformed parent phase. Figure 4.9 shows an arbitrary vector r (represented by the straight line OV) in the parent phase which becomes a zigzag line OA B C D U V in the twinned V

2 1

V′

U′

2 OV = r 1 2 C′

OV′ = r′ D′

1 (1–x )

2 A′ 1

B′ (x )

O

Figure 4.9. Schematic appearance of internally twinned martensite minor and major regions, which undergo the lattice deformation along different but crystallographically equivalent principal axes.

274


martensite crystal where the fractional thicknesses of the two constituent variants 1 and 2 are (1 − x) and x, respectively. The vector r is transformed into the vector r , the latter being an average of the segments OA , A B , B C U V . Thus r is the vector sum OV = OA + A B + B C + · · · · · + U V and can be expressed in terms of the pure lattice distortions and rigid body rotation as

or where

r = 1 − x1 B1 + x2 B2 r

(4.6)

r = Er

(4.7)

E = 1 − x1 B1 + x2 B2

(4.8)

The total distortion matrix, E, when it operates on any vector in the parent lattice, produces the corresponding vector in the transformed twinned martensite. Since the habit plane is an undistorted and unrotated plane, any vector lying on this plane will satisfy the following condition: Er = r

(4.9)

The rigid body rotations in the regions 1 and 2, as given by 1 and 2 , bring the planes (represented by AC and A2 C2 in Figure 4.8) derived from the mirror plane into coincidence. A rotation (2 = 1 ), which gives the relative rotation between 1 and 2 , can be defined and the total macroscopic shear can be expressed as E = 1 1 − xB1 + xB2 = 1 G

(4.10)

where G = 1 − xB1 + x B2 The macroscopic distortion thus has the following three components: 1 , a rigid body rotation: G, a fraction of the twin shear; and the Bain strain, B1 . The matrix algebra problem then reduces to an eigenvalue problem for vectors in the habit plane with solutions (if they exist) only for certain values of x, fraction of the twin shear (or the magnitude of the LIS). From a knowledge of the lattice parameters of and the lattice correspondence between the parent and the product lattices and of the twinning system, it is possible to predict the indices of the macroscopic habit plane, the orientation relationship and the twin thickness ratio, which are all experimental observables. The success of the phenomenological crystallographic theory is well documented in the extensive literature wherein the predicted habit planes in different systems have been shown to match closely with those experimentally determined (Table 4.1).


275

Table 4.1. Comparison of experimentally determined and theoretically computed crystallographic parameters of martensites in different systems. System

Habit plane Exp.

fcc–bcc Fe–30.9Ni

0.1656

Orientation relationship

Comp.

Exp.

0.1848

0.7998

0.7823

0.5771

0.5948

Comp.

(111)f ∧(011)b ¯ f ∧(011) ¯ b 1¯ 12

0.3

0.54

2.2

1.67

¯ f ∧ 1¯ 11 ¯ b 101

−24

−362

(111)f ∧(011)b ¯ f ∧(011) ¯ b 1¯ 12

0.86 ± 0.10

0.83

¯ f ∧ 1¯ 11 ¯ b 101

4.42 ± 0.10

−444

T for plates and L > B > T for laths. Apart from the differences in the relative external dimensions, the lath and the plate morphologies differ with respect to the nature of the assembly of the martensite units. While a plate morphology is characterized by groups of martensite units of differing orientation and habit plane variants, the distinctive feature of the lath morphology is the occurrence of groups of near parallel martensite units which are separated from each other by small angle boundaries. In some alloys, adjacent martensite laths exhibit a twin relation. The grouping of martensite units in some typical patterns is motivated essentially by their tendency towards self-accommodation. The overall strain energy of the system can be substantially reduced by an appropriate grouping of martensite units as is reflected in the formation of energetically favourable polydomain morphologies in several martensites. The issue of self-accommodation is of primary importance in shape memory alloys and will be addressed in detail in Section 4.5.5. 4.2.3 Kinetics Being a first-order-type transition, martensitic transformations occur by the nucleation and growth process. The overall kinetics of the transformation in most cases are athermal. This can be represented in a plot of fraction transformed versus temperature (Figure 4.10(a) and (b)). A transformation on cooling begins at the Ms

278


~100%

T1

Martensite (%)

Martensite (%)

Martensite (%)

100%

MS Mf

Mf

Temperature (a)

Ms > T1 > Mf

Mb

Temperature (b)

Time (c)

Figure 4.10. Kinetics of the martensite transformation as a plot of fraction transformed versus temperature for athermal martensite (a), athermal burst martensite (b) and overall transformation kinetics for isothermal martensite (c).

temperature; the extent of the transformation progressively increases with lowering of temperature and it is completed, finally attaining the complete transformation at the temperature Mf , which is known as the martensite finish temperature (Figure 4.10(a)). In some cases of athermal martensitic transformation, the volume fraction transformed at Ms shows a sharp rise in a burst, as shown in Figure 4.10(b); therefore, the start temperature is designated as burst temperature (Mb ). The time taken to reach the indicated fraction transformed at any given temperature between Ms and Mf is very short, and longer holding at the same temperature does not result in further increase in the fraction transformed. In another variety of martensitic transformations, overall isothermal characteristics are exhibited in the overall transformation kinetics (Figure 4.10(c)). In these cases, the volume fraction of the martensitic phase keeps on increasing with time at any given temperature between Ms and Mf . In such cases, although the growth of martensite units occurs by the characteristic athermal movement of glissile interfaces, the process of nucleation is thermally activated. Experimental studies on martensite nucleation have been carried out quite extensively in systems which show the isothermal behaviour. From such studies, it has been established that a homogeneous nucleation process through thermal activation cannot account for the observed nucleation at very low temperatures (even at temperatures approaching 0 K). Kaufman and Cohen (1956) invoked the presence of pre-existing martensite embryos to explain the observed nucleation phenomenon. The same anomaly is encountered if one considers the nucleation of martensite in an ideally perfect parent crystal. Assuming the shape of a martensite nucleus to be a thin oblate spheroid and choosing realistic values of the chemical free energy change due to the transformation, the surface energy


279

and the strain energy of the assembly of the nucleus and the matrix, the nucleation barrier (F ) can be estimated to be about 5 × 103 eV per nucleation event. This corresponds to about 105 kT at temperatures where nucleation is experimentally observed. This indicates that the thermal energy is much too small for homogeneous nucleation to occur. The postulation of pre-existing embryos which could act as heterogeneous sites for martensite nucleation was recognized to resolve this anomaly. In the early stages of the development of the theory of martensite nucleation, these embryos were conceived as being structurally similar to the martensite phase (such as bcc embryos in fcc austenite in ferrous alloys). However, no experimental evidence in support of the presence of such embryos could be obtained. Though there is a general concurrence on the requirement of heterogeneous nucleation in martensitic transformations, a precise structural description of the heterogeneities at which nucleation occurs is still not available. Olson and Cohen (1976) proposed a general mechanism of martensite nucleation by faulting of groups of existing dislocations. For the fcc → bcc transformation, martensite nucleation can be considered in terms of the splitting of a group of dislocations which form a parallel array, one above the other, in the parent phase. Movement of partial dislocations subsequent to dissociation produces a martensite nucleus bounded by a coherent interface. There is an alternative approach to seeking an answer to the problem of martensite nucleation. In a number of systems, martensitic transformations are preceded by precursor phenomena, usually known as “premartensitic” effects. Elastic moduli of the parent phase in several systems “soften” prior to the transformation. Clapp (1973) proposed a “strain spinodal” approach according to which a localized soft phonon mode may operate near a lattice defect, resulting in the nucleation of the martensite phase. The possibility of heterophase fluctuations aided by elastic interactions with pre-existing dislocations to produce martensite has also been considered. There have been only a few investigations in which the growth velocities of martensite plates have been experimentally measured. Two types of martensites have been encountered, one grows very rapidly and the other at a much slower pace; the former is termed “umklapp” and the later “schiebung”. From “in situ” monitoring of resistivity, the growth velocity of the martensite interface has been determined to be 1100 m/s for “umklapp” and 10−4 m/s for “schiebung”, respectively. The rapid movement of the martensitic interface is driven by the free energy difference between the parent and the martensitic phases. The interface can be visualized as being semicoherent in nature. The movement of the coherent segments is not opposed by any reactive force, while the movement of the interfacial dislocations involves Peierls stress due to

280


lattice friction and other internal stresses opposing their motion. This movement may be thermally activated if the interaction of the internal strain fields with the stress fields of dislocations is short range, but usually the internal friction stress acts as a long-range stress and, therefore, an athermal movement of the interface is necessary. The dislocations at the interface which acts as the transformation front have two functions: to accommodate the misfit between the lattices of the parent and the martensite phases and to generate the LIS by their propagation. In fact, there are three possible criteria for the selection of the interfacial array of mismatch dislocations: (a) the criterion of the minimum interfacial energy, (b) the criterion of the minimum force required to move the array and (c) the criterion of the fulfilment of the requirement of LIS. Since all the criteria cannot always be satisfied by the same set of dislocations, the third criterion is often chosen for modelling the martensite interface. The fact that in a majority of systems martensite interfaces propagate very rapidly in an athermal manner suggests that the relationship between the velocity of the interface and the force causing its movement (which is derived from the free energy difference between the parent and the product phases at the transformation temperature) contains an instability. It is likely that the instability is due to the fact that the driving force required to nucleate a martensite plate is much greater than that required for its growth. 4.2.4 Summary Various features of martensitic transformations have been briefly presented in the foregoing sections. The experimental observables on the basis of which one can identify a transformation to be martensitic have also been mentioned in the course of the presentation. Based on these considerations, the atomistic mechanism of martensitic transformations has been conceived to involve jumps of atoms from the parent lattice sites to the product lattice sites in a coordinated or disciplined manner by maintaining a lattice correspondence. If one could label the atoms along a vector in the parent lattice, one would observe that the same atoms occupy sites along a vector in the product lattice in the same sequence. In the same way, a labelled plane in the parent becomes a similarly labelled plane in the product. This is what, as illustrated in Figure 4.5, is known as a lattice correspondence. The maintenance of a lattice correspondence is possible only if atomic jumps from a specific lattice site of the parent to a corresponding lattice site of the product are predestined. This is evidenced from the fact that the chemical order of the parent phase is inherited by the martensite phase. Table 4.2 summarizes the distinguishing features of martensitic transformations.


281

Table 4.2. Characteristic features of martensitic transformation. 1. 2. 3. 4. 5. 6.

7.

4.3

Coordinated/disciplined jumps of atoms from parent lattice sites to product lattice sites Strict lattice correspondence between the parent and the product lattices Strict orientation relation between the parent and the product lattices Occurrence of surface tilts representing the macroscopic shears associated with martensite plates Inheritance of the chemical composition and the state of atomic ordering from the parent to the martensitic phase Transformation through a nucleation and growth process, the nucleation step being either athermal or thermally activated, while the growth process is invariably athermal. The growth of martensite plate by a rapid movement of a glissile coherent interface in a manner similar to propagation of a shear front

BCC TO ORTHOHEXAGONAL MARTENSITIC TRANSFORMATION IN ALLOYS BASED ON Ti AND Zr

Pure Ti and Zr transform martensitically from the high-temperature (bcc) phase to the low-temperature (hcp) phase on quenching from the -phase field, provided the cooling rate exceeds a certain critical value. There have been only a few experimental investigations of the critical cooling rate for martensitic transformation in these pure metals. In order to suppress the competing diffusional transformation (massive → transformation), a quenching rate of several hundred degrees celsius per second is necessary in the case of these metals (if the total interstitial content is less than 200 ppm). Once the massive → transformation is bypassed, the same structural change occurs through the martensitic process with the attainment of the required supercooling at the Ms temperature which is about 50 K lower than the equilibrium → transition temperature. The Ms temperature for alloys of Ti and Zr is a function of the alloy composition. The Ms temperature of any given alloy is determined by the - or the -stabilizing tendencies and the amounts of the alloying elements present in it. There are three possible athermal transformation products in -quenched dilute alloys of Ti and Zr. These are the hcp and the orthorhombic martensites and the athermal phase which has a hexagonal crystal structure. The mechanism of the → transformation is quite different from those of the → or the → martensitic transformations and has been discussed in a separate chapter. The orthorhombic martensitic phase, which forms in alloys containing large concentrations of -stabilizing elements, can as well be considered as a distorted hexagonal phase,

282


the orthorhombic distortion being introduced by a high level of supersaturation of alloying elements. The crystallography of the transformation can also be described in general terms for the bcc to orthorhombic structure, the latter encompassing the hcp structure as a special case. 4.3.1 Phase diagrams and Ms temperatures The range of alloy compositions over which the different athermal products, ,

and , form in a binary system based on Ti or Zr can be illustrated in a schematic isomorphous phase diagram (Figure 4.11). The Ms temperature for the martensitic transformation and the start temperature, s , as functions of xB , the atom fraction of a -stabilizing alloying element, are superimposed on this schematic phase diagram.

Ms

Tβ/α

Mf

T1

} for β → α′, α′′

Temperature

T2

ωs for β → ω

T4

α

T5

x1

Ti Zr

x2

x ′2

x3

xB Atom fraction of alloying element β-Quenched

α′

α″

β+ω

β

α″ + β + ω

α + β Alloy

Alloy classes

β Alloy

α Alloy

Figure 4.11. Basis for the classifications of commercial Ti and Zr alloys into alloys, + alloys and alloys.


283

On quenching from the -phase, alloys of different compositions exhibit different athermal transformation products. With reference to Figure 4.11, which is a modified version of Figure 1.18, alloys in the composition range 0 > xB > x1 produce (hcp) martensite, while those in the range x1 > xB > x2 transform into orthorhombic martensite, x1 defining the level of supersaturation at which orthorhombic distortion sets in. The plots corresponding to Mf and s as functions of xB intersect at xB = x2 . This implies that for alloy compositions xB < x2 , the martensitic transformation reaches completion during a quenching operation before the s temperature is encountered and therefore the product is fully martensitic. The temperature gap between Ms and Mf in most of the Ti- and Zr-based alloys is very small (in the range of 25 K, as reported in a few systems). Because of this reason, an incomplete martensitic transformation is not frequently observed. In the composition range of x2 < xB < x2 , the quenched structure consists of martensitic plates along with some untransformed -phase in which the -phase is finely distributed. In the composition range of x2 < xB < x3 , the quenched product contains a distribution of athermal -particles in the -matrix. The reason for the formation of a dual phase + structure in preference to a fully -structure will be discussed in the chapter on transformation. Quenching from the ( + ) phase field results in duplex microstructures. Depending on the temperature of equilibration in the ( + ) phase field, a wide variety of microstructures can be produced. This point can be illustrated by taking the example of the alloy composition, x4 . Table 4.3 lists the product phases in this alloy when quenched from the different temperatures, T1 , T2 , T3 and T4 as marked in Figure 4.11 which also shows the tie lines at these temperatures. Figure 4.11 could serve as a basis for the classification of commercial Ti alloys into , + and alloys. Since Zr alloys are almost exclusively used for structural applications in nuclear reactors, concentrated alloys do not find much use due to their high thermal neutron absorption cross-sections. Therefore, such a classification scheme is not in vogue for Zr alloys. However, from physical metallurgy considerations, Table 4.3. Microstructures produced in the alloy of composition x4 (as indicated in Figure 4.11) on quenching from different solutionizing temperatures. Solutionizing temperature

Microstructure

T1 T2 T3 T4 T5

Fully martensitic Primary + Primary + + Primary + + Primary +

284


the same classification scheme is also applicable for Zr alloys. The classification scheme is briefly described in the following. Alpha alloys are those which on equilibration at temperatures close to 873 K consist of the single-phase hcp ( ) structure. Alloys which on quenching from the phase field retain the -phase with or without a distribution of the -phase are grouped as alloys. The composition range between those of the alloys and the alloys is covered by the ( +) alloys. These three composition ranges are marked in the schematic phase diagram in Figure 4.11. The composition ranges corresponding to these three classes of alloys can be better defined in ternary alloys where the stabilities of the and the -phases are balanced by suitable alloying additions. The Ti-Al-V system can be chosen as a good example. The pseudobinary phase diagram (Figure 4.12) shows an enlarged -phase stability region due to the presence of the strong -stabilizing element Al. A ternary composition Ti–6% Al–3.5% V marks the limit of the -alloys, while a complete suppression of the → martensitic transformation on quenching requires an addition of about 15% V in a Ti alloy containing 6% Al. This composition (6% Al, 15% V) marks the low V limit for the -alloys. As indicated earlier, the composition range between 3.5 and 15% V corresponds to the + class of alloys. It may be noted that a -quenching treatment given to the ( + ) alloys produces a fully martensitic ( or ) structure. On subsequent tempering, the -phase (and/or the pertinent intermetallic phases) can be precipitated in the tempered martensitic matrix of these alloys.

4 wt% Al vertical section

Temperature (K)

1000

900

800

700

0

6.9Al 93.1Tl 0V

1

2

3

4

5

6

7

Atomic percent V

Figure 4.12. A pseudobinary diagram of Ti-Al-V system.

8

9 10

6.9Al 83.0Tl 10.1V


285

As mentioned in Section 4.2.1, thermodynamic studies on martensitic transformations essentially involve the determination of the relative stabilities of the parent and the product phases and of the To temperature at which the two phases of identical composition possess the same value of the integral molar free energy. Using the values of the free energy change, F → , associated with the to phase transformation for pure Zr and Ti, Kaufman (1959) determined the To line for the isomorphous binary Ti–Zr system. An outline of this thermodynamic treatment is given here to illustrate the special case where both the pure metals involved exhibit similar allotropic transformations and have complete solid solubility in both the high- and the low-temperature phases. The free energies for the and the phases, F and F , in Ti–Zr solid solutions can be expressed as F = 1 − xFTi − xFZr − Fex + RTx ln x − 1 − x ln1 − x

(4.13)

F = 1 − xFTi − xFZr − Fex + RTx ln x − 1 − x ln1 − x

(4.14)

and

where FTi Zr are free energies of the pure components and Fex are the excess free energies of mixing for the and the phases; x is the fraction of Zr in the Ti–Zr alloy. The condition for equilibrium between the two phases is given by the following identities at any given temperature T :

F Ti x = F Ti x

and

F Zr x = F Zr x

(4.15)

where F Ti x and F Ti x are partial molar free energies or chemical potentials of Ti in the solid solution of compositions x and x representing the compositions of the and the phases in equilibrium at T (represented by tie lines drawn on the phase diagram in Figure 4.13. Equations (4.11) – (4.13) yield 1 − x Fex Fex

→

FTi + RT ln x − Fex − x x = Fex − x (4.16) 1 − x x x and FZr → + RT

x Fex Fex

x − Fex + 1 − x x (4.17) ln = Fex + 1 − x x x

The martensitic transformation being composition invariant, the chemical driving force for the → transition for a Ti–Zr alloy of composition x is given by F → = 1 − xFTi → + xFZr → + Fex →

(4.18)

286

Phase Transformations: Titanium and Zirconium Alloys xβ

xβ

1100

β (bcc)

Temperature (K)

To (calculated) 1000

α+β

xα

xα

900

Ms (Duwez (1951))

800

α (hcp)

700 0

10

20

30

Ti

40

50

60

70

80

Atomic percent Zr

90

100

Zr

Figure 4.13. Ti–Zr phase diagram showing the position of Ms temperature for Ti–Zr alloys.

At the To temperature, F → = 0, while at the Ms temperature, the value of F → is adequate to provide the surface and the strain energies necessary for initiating the transformation. The free energy differences between the allotropes

and of pure Ti and Zr, expressed as FTi → and FZr → , respectively, have been evaluated in Section 4.1. In order to determine the excess free energies of mixing for the and the phases, Fex can be expressed in a power series expansion: Fex = x1 − xAo + A1 x + A2 x2 + A3 x3 + · · ·

(4.19)

where the coefficients Ai are temperature dependent. By choosing equilibrium conditions at different temperatures, which means substituting in Eqs. (4.14) and (4.15) the free energy values for different temperatures and the corresponding x and x from the phase diagram, several values of the coefficients Ai upto the nth order term can be determined in a generalized manner; this procedure is known as analysis of phase diagrams (Rudman 1970). In the case of the Ti–Zr system, Kaufman (1959) has shown that the approxima = x1 − xAo and Fex = x1 − xBo , tion of using only the zeroth order term, (Fex


287

can also yield reasonable values of To . With these approximations, Eqs. (4.14) and (4.15) are reduced to 1 − x = x 2 A − x 2 B 1 − x

(4.20)

x = 1 − x 2 A − 1 − x 2 B x

(4.21)

FTi → + RT ln and FZr → + RT ln

Since the Ti–Zr isomorphous phase diagram shows a minimum, at any given temperature of the / equilibrium, two tie lines and correspondingly two sets of x and x values, one on the Ti-rich side and the other on the Zr-rich side, can be obtained. Values of B − A, calculated from the Ti-rich and the Zr-rich sides, have been found to be in good agreement; these values can be expressed at temperatures between 1100 and 810 K: B − A = −2340 + 126T cal/mol

(4.22)

Consequently, Eq. (4.16) can be written explicitly for the Ti–Zr system as F → = 1 − xFTi → + xFZr → − x1 − x2340 − 126T cal/mol (4.23) Equation (4.21) can be used for calculating the To temperature as a function of x (the results are presented in Figure 4.13) and for computing F → as a function of T for different compositions of Ti–Zr alloys. Kaufman’s plots for F → versus T for different alloy compositions, superimposed with experimentally determined Ms temperature values (Figure 4.14), indicate that a chemical driving force of about 50 cal/mol is required for initiating the → transformation martensitically. This corresponds to a supercooling (To − Ms ) of about 50 K. A comparison of the chemical free energy requirements for martensitic transformations in alloys based on Ti and Zr with those in ferrous alloys indicates that this requirement is much smaller in the case of the former (50 cal/mol as against 300 cal/mol for ferrous alloys). This suggests that the restraining forces comprising strain energy and surface energy associated with the martensites in Ti and Zr alloys are considerably smaller compared to those associated with ferrous alloy martensites. This point will be dealt with while discussing the relative values of lattice strains. In binary phase diagrams involving Ti or Zr on one side and a -stabilizing element B (such as Mo, Nb, Ta or V) on the other, the Ti/Zr-rich side can be approximated as a isomorphous system. The chemical driving force for the

288

Phase Transformations: Titanium and Zirconium Alloys x = 0.2

x = 0.3

Titanium-rich

x = 0.1

ΔF α′ → β/ ~40 Cal/mol Ms

Difference in free energy ΔF α′→β (cal/mol)

+100

Ms

0

To –100

(ΔF

α′→β

x = 0.0

= 0) x = 0.4

x = 0.5 x = 0.8

Zirconium-rich

x = 1.0

x = 0.9

ΔF

α′→β

/ ~60 Cal/mol Ms

+100

Ms

0

To –100

(ΔF α′ → β = 0) x = 0.6

700

800

900

x = 0.7 1000

1100

1200

Temperature (K)

Figure 4.14. The chemical driving force for martensitic transformation in Ti–Zr alloys as a function of composition and temperature.

→ transformation in moderately dilute solutions (x < 007 x < 015 x < 020) can be approximately written (Kaufman and Cohen 1956, Kaufman 1959) as x x − 2x + x 2 1 − x

→

→

→ FTiZr + RT ln = 1 − xFTiZr − xRT ln − F x x 2 1 − x (4.24) The calculated To versus x plots and experimental Ms versus x plots (Duwez 1951, 1953) are superimposed on the phase diagrams of Ti–Mo and Ti–V systems in Figure 4.15. The fact that the Ms line lies 25–50 K below the To line is consistent with the general trend of the thermodynamics of martensitic transformations in these systems. Experimental values of Ms temperatures for various alloys based on Ti are plotted in Figure 4.15.

Martensitic Transformations Calculated To

1200

Ti – Ta 1100

β

Ti – W β

α+β

α+β

1000

α

To To

900

Ms Temperature (K)

289

Ms

800

Sotubility ≈ 0.2 at 973 K Ti – V

1200

1100

Ti – Nb

β

β

1000

To

α

900

α+β

α

α+β

To 800

Ms

Ms 0

5

10

15

0

5

10

15

20

Atomic percent alloying element

Figure 4.15. BCC and HCP phase relations in Ti-based alloys. Calculated To –x curve is compared with the observed Ms values.

4.3.2 Lattice correspondence The lattices of the parent and the product phases can be related in a number of ways. The correct choice of the lattice correspondence is generally made by selecting one which involves the minimum distortion and rotation of the lattice vectors. The choice made by Burgers (1934), as illustrated in Figure 4.16, shows that the basal plane of is derived from an {011} -type plane and that [011] and [100] directions transform into [0110] and [2110] directions, respectively. The close-packed directions [111] and [111] lying on the 110 plane transform to ¯ directions are derived two close-packed directions. The other from directions.

290

Phase Transformations: Titanium and Zirconium Alloys [011]β// [0001]α

(0001)α

[011]β// [0110]α [111]β// [1210]α

[111]β// [1120]α

[100]β// [2110]α

Figure 4.16. The distorted closed-packed hexagonal cell (hcp), derived from the parent bcc lattice.

As mentioned earlier, martensitic transformations in these systems result in the formation of either an hcp or an orthorhombic structure, the former √ being a special case of the latter structure with the ratio of lattice parameters b/a = 3. In general, the orthohexagonal axes system can be used for describing the martensite crystallography covering both → and → transformations. The lattice correspondences between the bcc and the hcp and between the bcc and the orthorhombic structures are depicted in Figure 4.17. Six crystallographically equivalent lattice correspondences between the orthohexagonal and the bcc lattices are described and labelled as variants 1–6 in Table 4.4. The lattice (Bain) distortion B associated with this transformation is given by ⎤ 1 0 0 B = ⎣ 0 2 0 ⎦ 0 0 3 ⎡

(4.25)

where 1 = 23 a /a 2 = a /a and 3 = 1/2 c /a The substitution of the lattice parameter values for pure Zr shows that the lattice strains are approximately 10% tensile, 10% compressive and 2% tensile,


291

[001]

C

[001]β

[010]β

c

[100]β

[111]β // [1210]α

[100]o

a a3 a1

[011]β// [0110]α

a2

[111]β // [1120]α [010]

Cubic (β lattice)

Figure 4.17. The lattice correspondence between bcc and orthohexagonal cells for the bcc → hcp transformation. The primitive hcp cell is defined by the vectors a1 , a2 and c and the orthohexagonal cell by a= a1 , b= a1 + 2a2 and c. The broken lines show the position of the bcc unit cell. Table 4.4. Correspondence between orthohexagonal and cubic cell (oRc ). Variant

[1 0 0]o

[0 1 0]o

[0 0 1]o

1 1 3 4 5 6

[1 [0 [0 [0 [0 [1

¯ c [0 1 1] [1¯ 1 0]c [1 1 0]c [1¯ 0 1]c [1 0 1 ]c [0 1 1]c

[0 [1 [1 [1 [0 [0

0 0 0 1 1 0

0]c ¯ c 1] ¯ c 1] 0]c 0]c 0]c

1 1 1¯ 0 1 1¯

1]c 0]c 0]c ¯ c 1] 1]c 1]c

respectively, along 1 , 2 and 3 directions. It may be noted that the lattice strains in this case very nearly satisfy the IPS condition. The deviation from the IPS condition arises only from the 2% tensile strain in the direction perpendicular to the basal plane.

292


An approximate analysis of the crystallography of the martensitic transformation

can be performed by neglecting the 2% strain in the direction along [011]

[0001] (Kelly and Groves 1970). The Bain distortion, Ba , then reduces to ⎡ ⎤ 09 0 0 Ba = ⎣ 0 11 0 ⎦ (4.26) 0 0 0 The construction of the strain ellipsoid for the corresponding distortion is illustrated in Figure 4.18, which shows that the two vectors OP and OQ have not been distorted by the Bain distortion but have been rotated from their initial positions OP and OQ, respectively. Since there is no distortion in the direction perpendicular to the plane of the paper, the pair of vertical planes containing the vectors OP and OQ also remain undistorted. This follows from the theorem that a plane remains undistorted if three non-collinear vectors lying in that plane remain unchanged in length or, in other words, if the lengths of two vectors lying in the plane together with their included angle remain unchanged. The total strain requires a rigid body rotation which rotates either OP or OQ to its earlier position. If one chooses the vertical plane passing through OQ as the habit plane, a clockwise rigid body rotation is required for bringing the undistorted

[110]β 10% Tensile

X

P′

X′

P

[011]β

10% Compression [011]β

O

Q Q′

Y Y′

Figure 4.18. Strain ellipsoid construction for bcc to hcp lattice deformation in Ti and Zr alloys.


293

plane to its original position. Let the coordinates of the point Q be (x y). Operation of the Bain distortion, Ba , brings the point Q to Q, the latter having the coordinates (0.9 x, 1.1 y). The condition

OQ = OQ

(4.27)

or x2 + y2 = 09 x2 + 11 y2 yields x/y = 105. The habit plan which contains OQ and [011] will be at an angle tan−1 105= 465 with the [100] direction. It is interesting to note that even in this case, where no inhomogeneous lattice invariant deformation has been introduced, the habit plane is not a rational plane. The choice of the other vertical plane passing through OP as the habit plane gives the second solution which is crystallographically equivalent to the former. This is a manifestation of the fact that the plane (100) , i.e. the vertical plane passing through [011] , is a mirror plane of the parent bcc structure. The next step is to determine the orientation relationship between the and the phases. The approximate Bain strain, Ba , and the rigid body rotation maintain the (0001) plane parallel to the (011) plane, while the rotation brings the [111] direction close to the [211] direction (within 1.5 ). The orientation relationship, therefore, can be described as (0001)

(011) ; [110

[111 This is widely known as the Burgers orientation relation which should be differentiated from the Burgers lattice correspondence given by Figure 4.16. It may also be noted that the Burgers orientation relation renders the (112) plane ¯ plane. This planar correspondence will be shown nearly parallel to the (1100) to be of great significance in deciding the / interface plane in diffusional transformations in Ti- and Zr-based alloys. Though the lattice strain along the 3 direction is non-zero (about 2% extension), for most of the → martensitic transformations studied in Ti- and Zr-based alloys, the predictions of the habit plane and of the orientation relation from this approximate analysis are not far from those experimentally observed. This is why the approximate analysis is quite instructive in arriving at a general understanding of the transformation geometry. In this context, the work of Bywater and Christian (1972) may be cited in which a suitable alloy, Ti–22% Ta, was chosen in which the lattice strain along the 3 direction is indeed zero. The absence of internal twins and of any dislocation substructure in the martensite plates in this alloy experimentally validated the absence of lattice invariant shear in this case. If the extension along the [110]

[0001] direction, which is present in the Bain distortion for most of these alloys, is not neglected, no plane remains undistorted on the application of the Bain distortion. An LIS then becomes a necessity to

294


make the total shear satisfy the IPS condition. The crystallographic analysis of a specific case is discussed in the following section as an illustrative example. 4.3.3 Crystallographic analysis The → transformation in a Zr alloy (Zr–2.5% Nb) is chosen as the illustrative example for demonstrating the steps of the crystallographic analysis. Lattice deformation: Using the Burgers correspondence and substituting the lattice parameter values of the and the phases in the Zr–2.5% Nb alloy (a = 03211 nm, c = 05115 nm and a = 03577 nm) in Eq. (4.23), the lattice deformation matrix can be expressed as

089768 00 00

109942 00

B01 =

00 (4.28)

00 00 101213 where B01 is the strain matrix of correspondence variant 1 of Table 4.5. This Bain distortion is on the basis of an axes system defined by the principal strain directions (x along [011]

[100]o , y along [011]

[010]o and z along [011]

[001]o directions). The same Bain distortion can be expressed in the axes system of the bcc crystal by using the similarity transformation for variant 1. ⎡ ⎤ 089768 00 00 105578 −004365 ⎦ Bc1 = O Rc B01 O Rc −1 = ⎣ 00 (4.29) 00 −004365 105578 Table 4.5. Bain strain matrices (Bc ) of bcc to hcp transformation in Zr–2.5 Nb alloy in the cubic basis for the six possible correspondence variants of the martensitic phase. Variant 1

2

3

Bain strain ⎡ 0 0 ⎣ 0 0 ⎡ 0 ⎣ 0 0 0 ⎡ − ⎣ − 0 0

= 089768; = 105578;

Variant ⎤ ⎦

4

⎤ ⎦ ⎤ 0 0 ⎦

= −004365.

5

6

Bain strain ⎤ ⎡ 0 ⎣ 0 0 ⎦ 0 ⎡ 0 − ⎣ 0 0 − 0 ⎡ 0 0 ⎣ 0 − 0 −

⎤ ⎦ ⎤ ⎦


295

In a similar manner, the Bain distortion for any other martensite variant can be determined, all on the basis of the axes system of the parent bcc lattice: the corresponding matrices are given in Table 4.5. The Bain distortion, when applied to a unit sphere of the parent phase, produces an ellipsoid. The intersection of the ellipsoid and the unit sphere defines the locus of the position vectors, r, which remain undistorted on the application of the Bain distortion. The undistorted vectors define the Bain cone which can be obtained from the condition

Br 2 = r 2

(4.30)

The initial and the final positions of the Bain cone for the variant 1 are shown in the stereogram in Figure 4.18. Lattice invariant shear: The next step in the crystallographic analysis is to identify the mode and the system of invariant deformation which can occur either by slip or by twinning. The lattice invariant deformation is determined in such a way that in combination with the Bain distortion it will maintain a plane of zero distortion. The system of a lattice invariant deformation (simple shear) can be defined by the shear plane normal, m, and the shear direction, l. For a simple shear the vectors which remain undistorted in length lie on two planes, K1 and K2 , as shown in Figure 4.19; K1 is the shear plane and K2 is the second undistorted plane which makes an angle of 90 ± with the shear plane, before and after the shear operation, being related to the magnitude of shear by the equation g = 2 tan

(4.31)

g

g /2 g /2

After shear

Before shear K2

α

Ko α

K 2′

K1

n1

Figure 4.19. Schematic showing that vectors lying on planes K1 and K2 remain unchanged in length after application of a simple shear.

296


For a habit plane solution to exist, it is necessary that the traces of the undistorted planes K1 and K2 (the plane K2 after the operation of the shear) must intersect the initial Bain cone, Bi . This is due to the fact that the vectors defined by the points of intersection remain undistorted on the operation of either the Bain distortion or the LIS. Therefore, they are not distorted by the total shear as well. The above criterion has been expressed as l /m restriction by Bilby and Crocker (1961). They have given the following two inequalities for examining whether a given system of shear qualifies for being an LIS in a given transformation: m21 1 − 22 1 − 32 + m22 1 − 32 1 − 12 + m23 1 − 12 1 − 22 ≤ 0

(4.32)

l12 12 1 − 22 1 − 32 + l22 22 1 − 32 1 − 12 + l32 32 1 − 12 1 − 22 ≤ 0 (4.33) where [m1 m2 m3 ] is the direction normal to the shear plane and [l1 l2 l3 ] is the shear direction defined with reference to the axes system defined by the directions of the principal stress components 1 , 2 and 3 . In case the LIS occurs by twinning, the two twin components necessarily maintain crystallographically equivalent lattice correspondences. This imposes additional criteria for the selection of the system of LIS. For type I twinning, the twinning plane K1 should be derived from a mirror plane of the parent crystal, while for type II twinning, the direction of the twinning shear 2T should be derived from a diad of the parent crystal. Based on the lattice correspondence for variant 1, it can be seen that some of the variants of {100} and {110} mirror planes are transformed into {1012} and {1011} planes which qualify to be twinning planes of the LIS as they satisfy the ‘l’ and the ‘m’ criteria (Table 4.6). Mackenzie and Bowles (1957) classified the transformation on the basis of the operating twinning system for the LIS, designating {1012} twinning for class A and {1011} twinning for class B transformations. It is also seen from Table 4.7 that some of the mirror plane variants of the parent crystal are transformed into mirror planes of the martensite crystal (e.g. (100) and (011) transform into (2110) and (0001) , respectively), and therefore, these planes cannot be twin planes of the martensite crystal. In case the LIS occurs by slip, there is no restriction that the shear plane has to be derived from a mirror plane of the parent crystal. Otte (1970) has examined the suitability of a large number of shear systems for the LIS and has identified, on the basis of low values of the magnitude of the required shear, the following as the most probable shear systems: {1101} < 2113> , {0110} < 2110> . Predictions of crystallographic theory: Let us first consider the bcc to orthohexagonal transformation in which the LIS occurs by twinning on the


297

Table 4.6. The result of Bilby and Crocker criterion (1 and m) for all the possible shear systems in the product hcp martensitic phase for correspondence variant 1. Sr. No.

1(a) 1(b) 1(c) 1(d) 1(e) 1(f) 2(a) 2(b) 2(c) 2(d) 2(e) 2(f) 3(a) 3(b) 3(c) 3(d) 4(a) 4(b) 4(c) 4(d) 5(a) 5(b) 5(c) 5(d) 6(a) 6(b) 6(c) 7(a) 7(b) 7(c) 8(a) 8(b) 8(c)

bcc

hcp

Direction

Plane

Direction

Plane

[111] [111] [111] [111] [111] [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [011] ¯ [011] ¯ [011] ¯ [011] [010] [010] [010] [010] [100] [100] [100] [100] [113] [113] [113] ¯ [113] ¯ [113] ¯ [113] ¯ [311] ¯ [311] ¯ [311]

¯ (110) ¯ (011) ¯ (101) (112) (121) (211) (011) ¯ (101) (110) (121) ¯ (112) ¯ (112) (011) (111) (211) (311) (101) (001) (100) (102) (011) (010) (001) (012) ¯ (110) ¯ (121) ¯ (121) (110) ¯ (211) (121) (011) (112) ¯ (121)

¯ [21¯ 13] ¯ [21¯ 13] ¯ [21¯ 13] ¯ [21¯ 13] ¯ [21¯ 13] ¯ ¯ [2113] ¯ [1210] ¯ [1210] ¯ [1210] ¯ [1210] ¯ [1210] ¯ [1210] ¯ (0110) ¯ (0110) ¯ (0110) ¯ (0110) ¯ [0111] ¯ [0111] ¯ [0111] ¯ [0111] ¯ [21¯ 10] ¯ ¯ [2110] ¯ [21¯ 10] ¯ [21¯ 10] ¯ [1213] ¯ [1213] ¯ [1213] ¯ [1¯ 123] ¯ [1¯ 123] [1-23] ¯ [1100] ¯ [1100] ¯ [1100]

¯ 1) ¯ (110 ¯ (0110) ¯ (101¯ 1) ¯ (1121) ¯ (1211) ¯ (2112) (0001) ¯ (101¯ 1) ¯ (1011) ¯ (1013) ¯ (1013) ¯ (1013) (0001) ¯ (21¯ 14) ¯ (21¯ 12) ¯ (3034) ¯ (1101) ¯ (0112) ¯ (21¯ 10) ¯ (2314) (0001) ¯ (0112) ¯ (0112) ¯ (0114) ¯ 1) ¯ (110 ¯ (112¯ 1) ¯ (1010) ¯ (1011) ¯ (1100) ¯ [1¯ 123] (0001) ¯ (1121) ¯ (1121)

l

m

Burgers vector of these dislocations and the alignment of their line vectors along the traces of 10¯l1 planes support this contention. The dislocation structure and the lath interface structure undergo significant alterations during cooling from the relatively high transformation temperature. It is for this reason that lath boundaries and the internal dislocation structure characteristic of the martensitic transformation are often not preserved well in the final microstructure.


313

Plate morphology. The plate morphology is characterized by martensite plates, parallel sided or lenticular, forming along all the habit plane variants and dividing the parent phase grains by a sequential partitioning process. The fractal nature of this morphology is evident from the self-similarity of assemblies of martensite plates forming in successive generations. The first-generation plates span/extend across the entire grain of the parent phase. The partitioned grain is then amenable to transformation in the next stage. As a consequence, the second-generation plates are shorter in length and further partition the parent grain. The next generation of plates form within these fragmented untransformed spaces and the process continues till the transformation is complete (in some cases pockets of the parent phase remain untransformed). The pattern generated by a group of martensite plates belonging to one generation remains essentially the same though the dimensions of the plates decrease in every successive generation, maintaining the self-similarity of the structure. Detailed crystallographic analysis has been primarily carried out on martensite plates belonging to the earlier generations mainly because they are large and well formed and, therefore, better suited for the analysis of orientation relations and habit plane indices. A survey of the substructure of primary martensite plates in ¯ twinning is frequently observed within these Ti and Zr alloys reveals that 1011 plates (Figure 4.26). However, a number of primary martensite plates remain only dislocated. Quantitative data on the frequency of twinning in primary plates are not available, but the incidence of twinning is seen to increase with a lowering of the Ms temperature. Martensite plates forming in later generations are found to be less frequently twinned. Such a tendency is presumably related to the fact that the later-generation plates form in a matrix which is substantially work hardened by the stresses resulting from the plastic accommodation of the earlier-generation plates.

(a)

(b)

Figure 4.26. Bright field TEM micrographs showing plate martensites: (a) internally twined (thick twins) martensites and (b) internally twined martensite where the minor twin fraction is very thin.

314


Hammond and Kelly (1969) were the first to point out that the internally twinned martensite plates in Ti–Mn alloys could be grouped into two classes. The first group was characterized by a large spacing of thick 10¯l1 twins. The thicknesses of the major and the minor orientations were sufficiently large to produce a clearly resolvable zigzag habit which was made up of habit segments of the two twin-related variants present within a single martensite plate. The ratio of the thicknesses of the minor and the major twins was seen to be 1:4, a value which closely matched with that predicted from the magnitude of the LIS calculated from the phenomenological theory. Similar zigzag habit has also been observed in several Ti and Zr alloy martensites where the thicknesses of the major and the minor twin orientations are large (minor twins thicker than 100 nm). The second group of plates was characterized by the presence of very thin twins stacked within the plates (Figure 4.26). The average value of the ratio of the thicknesses of the minor and the major twin variants was much smaller compared to that predicted from the phenomenological theory. This was suggestive of the fact that an additional component of the lattice invariant shear (possibly slip) was operative within these plates. An analysis of habit plane variants of the two types of internally twinned plates (containing thick and thin twins) indicates that there exists an important difference between them in terms of the orientations of their minor twin components. In Bowles–Mackenzie theory, four non-equivalent habit plane solutions, denoted by ( ± ±), are obtained in a class A transformation, where class A corresponds to the LIS on 1011 planes which are derived from the 110 -type mirror planes of the parent bcc phase. For the lattice correspondence given in Figure 4.16 and for the lattice invariant deformation on the (1101) plane, the orientation relations corresponding to the ( − +) and the ( + +) solutions, as represented in the stereogram in Figure 4.27, are obtained by a rigid body rotation around the [0001] direction in either the clockwise or the anticlockwise direction. The most important difference between these two solutions arises from the fact that the (1101) plane on which the LIS acts remains nearly parallel (an angular deviation of 2 ) to the (011) mirror plane for the class A ( − +) solution, whereas the (¯l101 twin plane gets separated from the (110) plane from which it is derived by an angle of about 11 for the class A ( + +) solution. This means that the twin plane remains nearly parallel to the mirror plane in the former case even after the application of the rigid body rotation. In such a situation both the twin components follow the Burgers orientation relation very closely. The growth of a twinned martensite plate involves the creation of successive twin components as the plate lengthens. As a particular variant grows, the combined effect of the rigid body rotation and the Bain strain (R.B) tends to establish an invariant plane between the parent and the growing martensite variant. The deviation of R.B from

Martensitic Transformations 100

111

1011

1101

111

111 1210

101

1120

011

011

0002

101

011 0110 0110 101 110

1011

101

110

1120

0210 111

0110 011

011 0002

011

101

0210

111 1210

1101

1011

110

0110

111

2110 100

2110

1120

315

110 1011

1101

1120 111

111 2110

100

100

2110

Figure 4.27. Stereographic projection showing the orientation relations corresponding to class A ( −+) and class A( ++) solution of Bowles and Mackenzie formulation. These two orientations are derived from the clockwise and anticlockwise rotations along [110]

11001 .

the IPS condition causes a strain build-up which gets periodically corrected by the creation of a second variant at regular intervals. In the case of a plate which belongs to the ( − +) solution, both the twin components follow the Burgers orientation relation very closely, and therefore, the R.B combination operating within each twin component nearly satisfies the IPS condition. As a consequence, these two twin components can grow to a significant extent (the thickness of the minor twin variant exceeding 100 nm), maintaining independent habit segments with the parent phase. In the ( + +) case, the twin plane (¯l101 does not remain parallel to the (110) mirror plane subsequent to the rigid body rotation. The major twin orientation in this case satisfies the Burgers orientation relation and the IPS condition quite closely. However, the minor twin variant ( 2 ) is significantly away from the Burgers orientation relation, the (0001) plane being 9 away from (011) . The IPS condition is, therefore, not obeyed even approximately within the minor twin variant. The growth of such a variant would cause a substantial build-up of strain, resulting from the deviation from the IPS condition. This would restrict the growth of the 2 component. The observed stack of very thin twins in the martensite plates belonging to the ( + +) solution lends support to this contention. Since the requirement of LIS is not met in these plates in which the twin thickness ratio is much smaller than the predicted value of 1:4, the operation of an additional component of LIS in the form of slip appears essential. A periodic array of < c + a > dislocations along a different variant of 10¯l1 planes within

2 components is suggestive of the additional lattice invariant deformation.

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Substructure resulting from atomic shuffles: The phenomenological crystallographic theory discussed so far takes into account the fact that the total macroscopic shape strain ratifies the IPS condition. The necessity of introducing a lattice invariant deformation in order to achieve this condition in a great majority of alloys has also been explained. The lattice invariant deformation can be manifested either by creating a stacking of internal twins or by repeated slipping at periodic intervals. The former results in the production of the internally twinned martensite, while in the latter debris of dislocations left behind are responsible for a dislocated substructure. The lattice (Bain) deformation shown in Figure 4.28(a)–(c) transforms the orthorhombic unit cell whose dimensions (a, b and c) match those of the product martensite. However, a Bain strain does not bring all the atoms into the right positions consistent with the hexagonal symmetry of the product phase. An additional atomic shuffle is required for achieving the right atomic positions. This point was taken into account in the Burgers model for the bcc → hcp transformation. As shown in Figure 4.28, the bcc → hcp transformation can be accomplished by a shear which converts the (011) plane into the (0001) plane and a shuffle of the atoms at O locations in the figure into the positions B or C for obtaining the hcp stacking. It is this choice in the direction of shuffle (either O → B or O → C) which allows the formation of domains with different stacking sequences. Figure 4.29(a) indicates the stacking sequence for a perfect hcp structure while Figure 4.29(b) shows the stacking sequence in a crystal containing two domains, corresponding to the two alternative shuffle directions. Atoms in the A layers remain unaltered while in Figure 4.29(b) a domain boundary is created by introducing different shuffles in the two halves of the crystal. Such a domain boundary is created ¯ partial dislocation. The fault in Figure 4.29(c) can be by the passage of a 13 1010 converted to that of Figure 4.29(b) by the removal of a C layer of atoms at the arrow, by ¯ + 1 [0001] → 1 [2023] ¯ insertion of a 21 [0001] partial. The resulting reaction 13 [1010] 2 6 indicates that the faults in Figure 4.29(c) are associated with a displacement vector ¯ . However, since the fault boundary is created by the random shuffling of of 16 2023 atoms, there is no necessity for partial dislocations to be present. A typical substructure of martensite laths and plates, primary as well as secondary, both dislocated and twinned, consists of fine striations within the laths and plates. Such a substructure was recorded by Ericksons et al. (1969), Hammond and Kelly (1970) and Knowles and Smith (1981a,b) in Ti–Mn and Ti–Cr binary alloys. Selected area diffraction from any martensite plate and both shows streaking which has been characterized by Banerjee and Muralidharan (1998) by detailed tilting experiments. It has been observed that the plane of diffracted intensity (which causes streaking in diffraction patterns along different zone axes) extends along a plane nearly perpendicular to the martensite habit plane (as shown in a stereographic projection in Figure 4.30. This observation is also consistent with


317

[111] (011)

(211)

70.53°

(a) A A

A

(0001)

A B

C

O

57°

A

A A

(b)

(c)

Figure 4.28. (a) A basic cell for the hcp structure outlined within five bcc unit cells, before the transformation. Shear along [111] transforms this cell to a hexagonal unit cell. The choice in the direction of shuffle to obtain the hexagonal structure. (b) Homogeneous shear transforms this cell into a hexagonal unit cell and (c) shows a choice (OB or OC) in the direction of a shuffle to obtain the hcp structure (after D. Banerjee and Muraleedharan, 1998).

the fact that the fine-scale substructure observed in Ti martensites shows striations approximately perpendicular to the habit plane Figure 4.31. Tilting experiments have also established from the invisibility criteria that the displacement vector asso¯ or 1 (2023]. ¯ This conclusion ciated with the observed striations is either 13 [1010] 6 has been arrived at on the basis of the observation that the striations are visible ¯ only with g = n 1100 + m0001 when n = 0. The micrographs in Figure 4.31

318

Phase Transformations: Titanium and Zirconium Alloys (a)

(b) Fault with 1 6 [2023] displacement vector A C A C Faulting created A by wrong shuffle B A B Sh A C ea [1 r b A 01 y C 0] A C B A B A

A B A B A B A B A

1 2

1 3

Ins er t [00 ion 01 of ]

hcp stacking

(c)

Figure 4.29. (a) The stacking along the [0001] direction in a hcp structure. The fault in the stacking ¯ shear. created by alternate choices of shuffle and (b) the fault created by a 1/3 [1010]

(100) (1120)

(1010)

(2110) Na

Nb (0110) (434) HABIT (111)

(011)

(1100) (011)

← plane of intensity

[1213]

(1210) [2113]

(0001) (011) [1123]

(1210) (111)

[4223] (011) (1100)

(0110)

(2110) (111)

(1010)

(1120) (100)

Figure 4.30. A stereographic projection of the Ti and the bcc-Ti phases in the Burgers relationship (open circles). The direction of the streaking measured from bright field micrographs with different zones and the line drawn through them indicates the plane of intensity distribution resulting in the streaking (after D. Banerjee and Muraleedharan, 1998).


319

Figure 4.31. Micrographs showing three-dimensional domains and their boundaries (after D. Banerjee and Muraleedharan, 1998).

also suggest that the substructure does not merely consist of planar faults which extend across the martensite plates and laths but is more closely associated with three-dimensional domains and their boundaries. The domain structure has a close similarity to the antiphase domain structure in ordered alloys though the substructure is seen in the martensitic -phase which is not ordered. The domain structure is not isotropic in the sense that the bounding surfaces of the domains are predominantly along basal planes with some segments being present along prism and pyramidal planes. Based on the observations made on the details of the substructure of Ti alloy martensites, Banerjee and Muralidharan (1998) came up with a schematic diagram (Figure 4.32) showing the domain structure in relation to a section of a martensite plate. The domains have been named as “stacking domains”, the displacement vector associated with the boundaries separating [00

01

(00

01

)

M

id

rib

]

(2

11

0) Habit pla

ne

[12

10

]

Figure 4.32. A schematic diagram of the domain structure in relation to a section of a martensite plate (after D. Banerjee and Muraleedharan, 1998).

320


adjacent domains being that of a stacking fault in the fundamental hcp structure (unlike a fault vector in the superlattice of an ordered structure). The presence of strain contrast in the images of the domain boundaries has also been established. The origin of domain formation in a martensitic structure can be considered in the following manner. When a thin sheet of martensite (of the order of a few unit cells in dimension) forms by the expansion of a shear loop which transforms a specific set of 110 planes into the 0001 planes, the required atomic shuffles for accomplishing the structural change occur in a random fashion. This process creates a given domain. It is not possible to ascertain whether the shuffle occurs at the advancing transformation front or within the product after the manifestation of the shape strain within that small volume. With further propagation of the shear front, contiguous domains form by randomly selecting one of the shuffle directions within a domain. Since there is no change in the nearest neighbours across a (0001) interface between two adjacent domains, the (0001) surfaces become the predominant bounding surfaces of these domains. Short segments of domain boundaries form along prismatic and near prismatic planes to close the domain volume as columns parallel to the basal plane, extending across the plate thickness (as shown in schematic drawing in Figure 4.32). 4.3.3.2 Transition in morphology and substructure In the literature on the martensite in Fe-based alloys, a number of terminologies have been used for describing the morphology and the substructure of the martensite product. Various characteristics, such as the nature of the assembly of martensite units which includes variant distribution and stacking pattern, the extent of self-accommodation, the nature of LIS and the transformation (Ms ) temperature are taken into account in grouping the martensite products and assigning suitable terminologies for describing them. Krauss and Marder (1971) have made a survey and have reported that there is a definite preference for the use of the terms: lath martensite and plate martensite for the two broad classes of martensites, the characteristic features of which are given in Table 4.10. These attributes which have been identified essentially from a large body of experimental data on Fe-based martensites can as well be adopted for classifying martensites in Ti- and Zr-based alloys. As pointed out earlier, martensites in these alloys can be broadly grouped into the lath and the plate types. The facts that based on the same characteristic features, martensites in Ti- and Zr-based alloys can be classified into these two groups and that a transition is morphology and substructure can be introduced by suitably choosing the alloy composition suggest that there is something fundamental in this morphological transition. The crystallography of the transformations, fcc → bcc or bct in ferrous alloys and bcc → hcp or orthorhombic in Ti and Zr alloys, being quite distinct, there must be some common issues (not related to


321

Table 4.10. Characteristic features of two classes of martensitic products in Fe-based alloys. Attributes

Lath martensite

Plate martensite

Shape of the martensite unit, length, width and thickness given by a, b and c

a > b > c, parallel sided

a ≈ b c, acicular or lensshaped

Nature of stacking of of martensite units

A group of laths of nearly identical habit plane stacked in a parallel array within a packet: (a) martensite variants, (b) close orientations in a group and (c) alternate laths of twin-related variants

Multiple variants present along intersecting habit planes in the same microscopic domain

Mesoscopic morphology

Packets of martensite plates, several packets filling up the entire austenite grain

Fractal morphology arising from repeated partitioning of the austenite grains by martensite plates of different habit variants – smaller martensite plates filling up the partitioned austenite volume created by martensite plates of previous generation

Internal structure arising from the lattice invariant shear

Generally dislocated martensite (only in a limited cases twinning is observed within lath martensites)

Frequently encountered internally twinned plates

Transformation temperature and sequence of transformation events

Relatively high Ms temperature dilute alloys

Relative low Ms temperature Move concentrated alloys

Kinetics

Slowly growing martensite volume fraction with increasing cooling

Rapidly growing martensite volume fraction with increasing cooling

Mechanism

Martensite laths of successive generations weakly strain-coupled

Strong strain coupling of martensites of successive generations

Type

Schiebung martensite

Umklapp martensite

the crystallographic symmetries and the magnitude of the (Bain strain) which dictate the transition in morphology and substructure of martensites. The factors which appear to have strong influences on the morphology and substructure are (a) transformation temperature, (b) self-accommodation of a group of martensite

322


plates (c) plastic flow of the soft parent phase in partially accommodating the transformation strain and (d) autocatalytic nucleation of martensite plates. These factors are again interdependent. With decreasing Ms temperature, the flow stress of both the high-temperature parent phase and the low-temperature martensitic phase increases substantially. This enhances the strain energy accumulation due to the formation of a single martensite plate. The requirement of self-accommodation, therefore, becomes more important as the Ms temperature is lowered. The morphological transition from the lath to the plate type in several ferrous alloys is accompanied by a change in substructure from dislocated to twinned. The coincidence of morphological transition (lath to plate) and substructural transition (dislocated to twinned) has also been noticed in Ti- and Zr-based martensites (Banerjee and Krishnan 1973). It is still difficult to conclude whether the transition in one is instrumental in bringing about the other transition. Figure 4.33(a) shows the Ms temperature as a function of alloy content in several binary alloys of Ti and Zr. It is seen that as the Ms temperature is brought down below about 923 K, both morphological and substructural transitions occur. In fact, the correlation between the transition in morphology and substructure and the Ms temperature is universally valid for Ti and Zr systems whereas the correlation is not so perfect in case of ferrous alloys. Let us now examine the plausible reasons for a transition in morphology and in substructure with lowering of the Ms temperature. When a martensite lath forms at sufficiently high temperatures where the matrix -phase remains relatively soft, the shape strain associated with the formation of a lath is primarily accommodated by the plastic flow of the matrix. No significant stress field is generated around the first-formed laths and therefore, stress-assisted nucleation of other martensite variants around the first-generation laths remains quite restricted. The surfaces provided by the first-formed laths, however, act as favourable nucleation sites for new generation laths. Those orientation variants are favoured which have a habit plane close to that of the first-formed laths and whose orientations are either close to or twin related with the former. These tendencies are responsible for creating a packet of parallely stacked laths. As far as the selection of the LIS is concerned, a lower critical resolved shear stress for slip compared to that for twinning promotes the operation of the slip mode at high temperatures. This point is illustrated in Figure 4.33(b). With the lowering of the Ms temperature, the stress field created by the firstgeneration plates assists the nucleation of several martensite variants in the vicinity of the former. However, the principle of self-accommodation allows the nucleation and growth of those plates which permit minimization of the stress field associated with the assembly of plates. The grouping of variants is invariably controlled by


323

900

Lath-dislocated martensite

Temperature (°C)

800 700 600

Zr

Ti

Ta

500

Cu

Nb

Cr

400

Acicular (plate) twinned martensite

300 200 Zr

5

10

15

20

20

15

10

5

Ti

Atom % (a) Slipped

Stress

Twinned

Twinning

Slip Temperature (b)

Figure 4.33. (a) Plots of the Ms temperature versus the alloy content in several binary alloys of Zr and Ti showing the necessary level of alloy additions to bring about a change in the morphology and the substructure. (b) A schematic diagram illustrating the variation of the CRSS for slip and for twinning as a function of temperature. The diagram suggests that a lowering of the Ms temperature would result in a slip to twin cross-over in the operating mode of inhomogeneous shear.

the principle of self-accommodation, leading to a polydomain morphology with least stored elastic energy. The elastic bonding of one plate with its neighbouring plates is the primary driving force for the generation of the plate morphology. Low transformation temperatures also promote the occurrence of twinning which predominates at least in the primary martensite plates in plate morphology.

324


4.3.4 Stress-assisted and strain-induced martensitic transformation The chemical free energy change which acts as the driving force for initiating a martensitic transformation can be supplemented by applied stress and/or plastic strain. As has been mentioned earlier, the Ms temperature defines the extent of supercooling (To −Ms ) which is required for the provision of the adequate chemical free energy change which can meet the surface and the strain energy requirements for martensite nucleation. It is because of this reason that the martensite start temperature is raised above Ms when external stress or plastic deformation is introduced in the parent phase. The interrelationships among applied stress, plastic strain and martensitic transformation have been extensively studied in Fe-Ni-C and Fe-Ni-Cr-C alloys by Bolling and Richman (1970a,b). They have defined a temperature, Ms (lying above Ms ) below which plastic yielding under applied stress is initiated by the onset of martensitic transformation and above which plastic flow of the parent phase precedes the nucleation of martensite. Olson and Cohen (1972) have provided a schematic representation of the interrelationships between applied stress, plastic strain of austensite and the stress required for martensitic nucleation (Figure 4.3) at temperatures above Ms . In the temperature range Ms < T < Ms , the stress required for martensite nucleation increases with temperature and follows a temperature dependence consistent with the elastic stress requirement for martensite nucleation; the phenomenon in this temperature range is, therefore, described as stress-assisted martensite nucleation. With increase in temperature within this range Ms < T < Ms , the stress requirement increases in a linear fashion as the driving chemical free energy change decreases approximately linearly with temperature. In the temperature range Ms < T < Md , the stress required for stress-assisted nucleation exceeds the yield strength of austenite, and therefore, the austenite initially yields by slip, causing extensive dislocation activity. With the accumulation of plastic strain (and dislocation multiplication), localized strain centres are created where martensite nucleation becomes energetically feasible. The yield strength of austenite marks the lower boundary of the stress required for strain-induced nucleation, as shown in Figure 4.3. To summarize the overall situation, the chemical driving force is sufficient at Ms for the pre-existing nucleation sites or embryos in the austenite to become operative without the application of stress. At temperatures between Ms and Ms , in the stress-assisted nucleation regime, such nucleation can occur only with the aid of applied stress. Here the required initiating stress is in the elastic range but increases with increasing temperature because of the concomitant decrease in the chemical driving force. At temperatures above Ms , plastic straining of the austenite results in the creation of martensite nuclei in the regime of straininduced nucleation. With increasing temperature above Ms , further reduction in


325

chemical driving force necessitates increased plastic strain in order to produce detectable amounts of transformation. This requires the initiating stress to rise above the yield stress of the austenite. As a result, plastic flow of the austenite, leading to extensive dislocation multiplication, occurs during the onset of plastic deformation. The nucleation of martensite becomes favourable at locations where strain accumulations exceed a certain threshold value. The formation of stress-assisted martensite is reflected in the appearance of a deviation from the linear elastic behaviour at a stress value lower than the yield stress value of the parent phase. Figure 4.34 illustrates the stress–strain plot for a Ti alloy in which stress-assisted martensite formation commenced at the point marked by an arrow. As mentioned earlier, martensite plates were introduced by applying external stress in alloys of suitable compositions, e.g. Ti–12.5% Mo (Gaunt and Christian 1959), Ti-V (Menon et al. 1980), in which the -phase could be retained in a metastable state by quenching. In such alloys, the Ms temperature in invariably below the room temperature and the martensite formed is primarily of the internally twinned plate type. TEM examination of these martensites has revealed the presence of periodic twins within these martensite plates (Figure 4.35). Since these martensites contain a fairly high level of -stabilizers (in order to lower the Ms temperature below room temperature), they often exhibit orthorhombic distortion, though it is not essential that all stress-assisted or strain-induced martensites

Engineering stress (MPa)

1000

800

FC

WQ

600

400

↓

200

+α –β heat-treated β heat-treated 0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Engineering strain

Figure 4.34. Stress–strain plot for a Ti alloy in which stress–assisted martensite formation commenced at the point marked by an arrow. While furnace cooled (FC) samples do not exhibit stressinduced martensite formation, water quenched (WQ) samples show deviation from linear elastic deformation at a low stress value (pointed by an arrow) corresponding to the formation of stressinduced martensite.

326


200 nm

200 nm

(a)

(b)

OM

β

OM

OM

750 nm

(c)

Figure 4.35. TEM micrographs showing martensitic plates with periodic twins.

in Ti- and Zr-based systems will be orthorhombic in nature. It may also be noted that stress-assisted martensite has been encountered only in a limited number of Zr-based alloys, the Zr-Mo-Al alloy system being an example in which clear evidence of stress-induced martensite has been reported.

4.4

STRENGTHENING DUE TO MARTENSITIC TRANSFORMATION

It is now well established that the strengthening due to the martensitic transformation arises from various factors such as solid solution hardening, precipitation hardening, order hardening and hardening due to lattice defects and fine crystallite size. Several attempts have been made to determine quantitatively the individual contributions of these factors towards the overall strengthening of martensites. Excellent reviews by Christian (1971) on ferrous martensites and by Warlimont and Delaye (1974) on non-ferrous martensites treat the subject in a comprehensive manner. Extensive research on the strength of ferrous martensites has demonstrated


327

that the major contribution to the strength of Fe–C martensites arises from the solid solution strengthening brought about by C atoms in the supersaturated body centred tetragonal phase and that the defect structure introduced by the martensitic transformation is responsible only for a relatively small increase in the strength level. That part of the strength which arises from solution hardening could be imparted to the base metal by introducing C atoms in excess of the solubility limit even by some alternative methods such as sputter deposition (Dahlgren and Merz, 1971) and thus cannot be attributed to the martensitic transformation per se. In fact, the strengthening due to a fine-scale precipitation or due to Snoek ordering of C atoms is also essentially connected with the C supersaturation in the martensite. Apart from solid solution hardening, a substantial part of the strength of martensites in several Cu-based alloys is derived from the ordering of substitutional atoms. In other words, the inheritance of a high level of solute concentration or an ordered arrangement of atoms from the parent phase is primarily responsible for the strengthening of martensites in Fe- and Cu-based alloys. Heat treatments involving the martensitic transformation have been found to be quite effective in strengthening Ti- and Zr-base alloys. The + alloys in these systems are amenable to the formation of the martensitic phase on quenching from the phase field. Standard heat treatments of these alloys involve a quenching treatment followed by a tempering treatment which results in the best combination of strength, ductility and fracture toughness. Unlike in the case of the Fe–C system, the hcp martensites in Ti- and Zr-based alloys are neither supersaturated with interstitial elements nor possess any distorted crystal structure. It is, therefore, interesting to examine the extents to which different factors are responsible in imparting strength to Ti and Zr alloy martensites. Banerjee et al. (1978) have investigated the strengthening mechanisms operative in martensites in binary Ti–Zr alloys. The isomorphous Ti–Zr system provides a unique opportunity in which the strengthening contributions of the solid solution and the martensite substructure can be separated. While quenching produces the martensitic structure, a slow cooling from the -phase field results in the formation of a Widmanstatten structure. The chemical composition (including the interstitial content) and the crystal structure corresponding to the and phases so formed being identical, the difference between the flow stress values of

and can be considered to arise from the differences in the defect structures of these products. The martensitic structure contains more closely spaced interfaces separating adjacent laths, plates and twins and a higher density of dislocations as compared to the slow-cooled Widmanstatten structure of the same composition. The as well as the phases in this isomorphous system possess the equilibrium composition. Since there is no tendency of phase separation in either of these phases, the possibility of fine-scale precipitation, usually pertinent to the Fe–C

328


martensite, can be ruled out. The → transformation attains completion both during the quenching and the slow cooling operations, and thus no interference from the retained parent phase is present. A comparison of the true stress versus true strain curves associated with the and structures of a given alloy directly yields the strengthening contribution, , which can be attributed solely to the martensitic substructure, being defined as the difference between the flow stresses of the and the structures, = − x , for a given value of the true plastic strain, p . The assumption which is implicit in this treatment is that the contributions of the various pertinent factors to the strength are additive. In order to quantify the microstructural difference between the Widmanstatten

and the martensitic (either dislocated lath or twinned plate morphology) phases, the average spacing between the partitioning interfaces in these has been measured for binary Zr–Ti alloys of different compositions (Table 4.11). While in the case of the Widmanstatten structure, the average thickness of the plates gives the mean free spacing between the partitioning interfaces, for the lath martensite structure, it is the average thickness of the laths and for the internally twinned plate martensite, it is the average twin thickness. The average packet size for the lath martensite structure has also been recorded as the packet boundaries are the effective barriers against the passage of slip or twin deformation. This point is discussed in a later section. The value of , which is a measure of the strengthening contribution solely of the martensitic structure, has been found to be strongly dependent on the alloy composition as seen in Figure 4.36. This is not unexpected in view of the fact that the morphology and substructure undergo a drastic change from the dislocated lath to the twinned plate martensite as the Ti content exceeds a limiting value of about 15% in Zr–Ti alloys. The sharp change in the value at around this composition strongly indicates that the twinned plate martensite structure provides a more effective barrier against dislocation motion. The value has also been plotted against the true plastic strain (Figure 4.37) for different alloy compositions. The sharp rise in with increasing plastic strain in the case of twinned plate Table 4.11. Average size of the and the units in Zr–Ti alloys. Average width of Widmanstatten plates ((m) Zr–5Ti Zr–10Ti Zr–15Ti Zr–20Ti

81 11.5 3.1 18

Average spacing between partitioning interfaces ((m) 0.8 0.4 0.2 0.15

Average packet size in the lath structure ((m) 60 25 – –


329

0.010 40

0.005 0.015 and 0.020

400

0.002

36

Transition in morphology and substructure lath → plate 32 dislocated → twin

300

24 200

20

MN/m2

Δσ (kg/mn2)

28

16 12 100 8 4 0

Zr

10

20

30

Ti atom percent

Figure 4.36. The rapid rise in corresponds to the composition range over which the martensite structure changed from the dislocated lath to the twinned plate type. The values of the plastic strain at which flow stress values were measured are indicated.

martensites suggests that this substructure exerts a powerful influence on the work hardening process. In contrast, the work hardening rate remains somewhat less sensitive to the plastic strain for dislocated lath martensites in pure Zr and in Zr–5% Ti and Zr–10% Ti alloys. 4.4.1 Microscopic interactions We will now examine the effectiveness of different types of interfaces in offering resistance to an approaching deformation front (either slip or twin). 4.4.1.1 Lath boundaries It has been observed (Banerjee et al. 1978) that a slip band can propagate through the small angle boundaries separating adjacent laths contained within a packet. Slip deformation can be induced in a thin foil sample by beam heating in the TEM. A dynamic experiment has shown that a set of dislocations can easily penetrate through the lath boundaries, the path followed by each dislocation being marked by a pair of slip traces which suffers a small angular deviation at the lath boundary. The stress required to push a dislocation out of a small angle boundary, made up of an array of parallel edge dislocations lying on a plane normal to the slip

330


700

σ (α′) σμ (α′)

400

σ (α)

300

σ∗ (α′)

100

0.02

0.02

σμ (α′)

500

Stress (MN/m2)

Stress (MN/m2)

500

0

0.02

σμ (α)

300

σμ (α)

200

σ∗ (α)

100

0.02

σ (α)

400

σ∗ (α′)

0

0.02

0

0.02

0.04

(a) 900

Zr –15% Ti

σ (α′)

σ (α)

500

σμ (α) 400

Stress (MN/m2)

600

Stress (MN/m2)

0.08

Zr –20% Ti

800 700

σ (α′)

600

σμ (α′)

σμ (α′)

700

0.06

(b) 900

800

σ∗ (α)

Plastic strain

Plastic strain

500

σ (α)

400

σμ (α) 300

300

200

200

σ∗ (α′)

σ∗ (α)

100 0

Zr –10% Ti

600

600

200

σ (α′)

Zr – 5% Ti

0.02

0.04

0.06

0.08

σ∗ (α′)

σ∗ (α)

100 0

0.0

0.02

0.04

Plastic strain

Plastic strain

(c)

(d)

0.06

0.08

Figure 4.37. The value plotted against the true plastic strain for different alloy compositions.


331

direction, x, is given by xy =

035b 2h1 − $

(4.53)

where h is the spacing between the dislocations, is the shear modulus and $ is the Poisson ratio. The stress xy can be generated at the head of a pile-up of n dislocations where n = xy , being the applied stress. Since n, and L, the average lath thickness, are related by #L b

(4.54)

0352 b2 2#Lh1 − $

(4.55)

n= Eq. (4.51) reduces to )2 =

Substituting typical values for L= 1m, h (= 40 nm) and b (= 032 nm), the magnitude of the applied stress, , required to push a dislocation out of the lath boundary can be estimated to be about /5000. Such a small value of justifies the easy penetration of glide dislocations through the lath boundaries. TEM of deformed lath martensites has also shown that deformation twins too can propagate through the lath boundaries which separate laths of nearly identical orientations. The packet boundaries separating laths of two distinct orientation variants have been found to be quite effective in stopping slip and deformation twin bands. It is, therefore, the packet size which is more important in deciding the strengthening contribution of the martensitic structure in dislocated lath martensites in Ti- and Zr-based alloys. Since the magnitude of the LIS is rather small in these alloys, the density of dislocations in these martensites is not very large. A high Ms temperature (above 923 K) is also responsible for a partial annihilation and rearrangement of dislocations resulting the LIS and the plastic accommodation. As a result, the strengthening contribution of dislocations in the lath martensite structure is not very significant. 4.4.1.2 Twin boundaries and plate boundaries A large proportion of the partitioning interfaces in the internally twinned plate martensite structure is constituted of 10¯I1 twin interfaces. Some of these are boundaries separating adjacent twins within single plates while some others are located between contiguous martensite plates obeying twin-related variants of the orientation relation.

332


Contrary to expectations, the increase in strength associated with the transition from a dislocated lath to a finely twinned substructure in steel martensites is not very large. Had the twin interfaces in steel martensites acted as effective barriers against dislocation motion, the internally twinned plate morphology would have been about three times as strong as lath martensites, consistent with an order of magnitude refinement of the mean free slip length. Experiments have shown twinned martensites to be only 8–30% stronger than lath martensites in ferrous systems. This discrepancy has been explained by Kelly and Pollard (1969) on the basis of the Sleeswyk–Verbraak mechanism (1961) of the passage of a/2 slip dislocations through the coherent twin interfaces. The effectiveness of 1011 -type twin interfaces in resisting the motion of dislocations in a twinned martensite plate in a Ti or a Zr alloy can be evaluated ¯ by considering the interaction of the different variants of slip in the 1010 ¯ < 1210 > system with a specific 1I01 twin interface. Yoo (1969) has expressed the dislocation reaction at a twin interface in a general form as Xhklm X∗ hklt ± nbt

(4.56)

where X is the Burgers vector of a dislocation gliding on the (hkl)m slip plane of the matrix which, after encountering a twin interface, gets converted to a new dislocation in the twin crystal with a Burgers vector X∗ (hkl)t which can glide on the (hkl)t plane of the twin orientation. While crossing the twin boundary, the dislocation creates a twin dislocation with Burgers vector nbt where bt is the Burgers vector of a unit twin dislocation. Using the relevant transformation matrices for transforming the Burgers vectors and Miller indices of the corresponding slip systems, the interaction of three variants of the first-order prismatic slip, {101¯ 0} , with a specific (11¯ 01) twin interface can be described by the following dislocation reactions: 1/32110m 0110m → 1/25146t 1343t − bt

(4.57)

1/31120m 1100m → 1/31120t 1103t

(4.58)

1/31210m 1010m → 1/21546t 3143t + bt

(4.59)

Out of these possible interactions, only in the second case can a dislocation with ¯ propagate through the (11¯ 01) twin interface without changing its b = 1/3 [1120] Burgers vector. The line of intersection of the (1¯ 100) slip plane and the (11¯ 01) ¯ m direction and, therefore, the dislocation line assumes twin plane defines the [1120] a perfect screw character as it encounters the twin plane; and as a consequence of this, it is amenable for cross slip on the (1¯ 01)t plane of the twin crystal. This


333

mechanism has been illustrated schematically in Figure 4.38(a). Passage of slip deformation in the other two cases is quite unlikely as the indices of the slip plane and direction are irrational and a twinning dislocation is left behind at the interface. Thus for these two slip systems, the (11¯ 01) twin plane acts as totally opaque. If one considers that the orientations of martensite crystals are nearly randomly distributed, the probability of the twin interface being completely opaque towards a first-order prismatic slip is 2/3. For the remaining 1/3 probability, the entry of 1/3 [1120] dislocations through a cross slip mechanism (illustrated in

(1101) Twin plane

ξ

b (1100)M

(1100)T

(a) (1122)

21°

3]

[112

(0112)

(1011)

1]

[011

(1011)

(2111)T

(0112)T

(b)

Figure 4.38. (a) A schematic showing the mechanism for cross slip on the (1¯ 01)t plane of the twin crystal (b) Schematics showing the entry of 1/3[1120] dislocations through a cross slip mechanism.

334


Figure 4.38(b)) is possible. However, the stress required to make the dislocation glide on the new slip plane in the twin segment would be different, depending on the Schmidt factors for the slip systems in the matrix and the twin segments. The analysis of the interaction of the most favourable prismatic slip mode with {101¯ 1} twin interfaces shows that for a majority (2/3 probability) of the orientations of the stress axis a single twin segment would behave as a plateshaped crystal strongly bounded by a pair of non-deformable interfaces such that neither can the slip dislocations penetrate these nor can there be any sliding along the coherent twin interface. The deformation of a thin crystal of thickness in the range of 50–200 nm, bounded by non-deformable twin interfaces, results in the generation of a strain gradient and as a consequence an array of geometrically necessary dislocations is created. Ashby and Johnson (1969) has shown that during the deformation of plastically inhomogeneous materials, geometrically necessary dislocations are produced and the dislocation density is proportional to /bL where and b are the magnitudes of the shear and the Burgers vector, respectively, and L is the spacing between the non-deformable interfaces. Banerjee et al. (1978) have shown that geometrically necessary dislocations indeed accumulate in Zr–Ti martensites subjected to plastic deformation. These dislocations are seen to be lined up along the twin boundaries and their density increases with lowering of the mean free slip length, L. The possibility of the passage of deformation twins cutting across transformation twins needs be considered in these hcp alloys in which different deformation twin systems are operative, particularly at low temperatures and at high strain rates. The most common deformation twin systems in these alloys are {101¯ 2} ¯ . The possibility of the , {1121} < 2113> and {1122} intersection of different variants of these deformation twins across a stack of (101¯ 1) transformation twins can be examined by the invoking the following plastic compatibility conditions proposed by Cahn (1953): (a) the traces of the crossing twin and the secondary twin in the composition (K1 ) plane of the crossed twin should be parallel and (b) the direction, the magnitude and the sense of the shear should be identical in the crossing and the crossed twins. If these plastic compatibility conditions are fulfilled, a deformation twin cannot be impeded by twin interfaces. From such an analysis, it can be seen that the following twin shears can pass through a set of (101¯ 1) transformation twins: (011¯ 2) [01¯ 11] , (11¯ 02) [1¯ 101] , (1122) [1123] and (21¯ 1¯ 2) [21¯ 1¯ 3] These types of intersections have been reported in the microstructure of the deformed martensite in Zr–15% Ti and Zr–20% Ti alloys (Banerjee et al. 1978). Figure 4.39 illustrates a few cases of intersection of deformation twins and transformation twins.


(a)

335

(b)

(c)

Figure 4.39. Examples of intersection of deformation twins and transformation twins.

4.4.2 Macroscopic flow behaviour The influence of partitioning interfaces, which are mostly impenetrable to dislocations, on the macroscopic flow behaviour of Zr–Ti alloys with respect to initial yielding as well as work hardening has been examined in this section. The accumulation of geometrically necessary dislocations in the vicinity of partitioning interfaces, observed in deformed Zr–Ti martensites, points to the fact that the flow behaviour of martensites can be rationalized in terms of Ashby’s one parameter work hardening theory for plastically inhomogeneous materials. According to this theory, the work hardening associated with a given microstructure characterized by a mean free slip length, , is expected to be governed by geometrically necessary dislocations if the density of these is much larger than that of statistically stored dislocations. Under such circumstances, the stress–strain plot would assume a parabolic shape given by the following relation: 1/2 b 1/2 (4.60) = 0 + C1 EM

where 0 and are the tensile stresses at strain levels zero and , respectively, C1 is a dimensionless constant which has been estimated to be 0.35 ± 0.15 for

336


plate-like barriers (Ashby, 1971), b is the Burgers vector, M is Taylor’s orientation factor and E is the Young’s modulus of a random polycrystalline sample. The applicability of Ashby’s one parameter work hardening theory to the plastic flow behaviour of Zr–Ti martensites has been demonstrated by the linear nature of the plots of versus 1/2 (Figure 4.40(a)). It can be seen that each flow curve can be fitted with two straight line segments, the first segment corresponding to strain values less than about 1% and showing a much larger work hardening rate than Zr – 15% Ti(α′) 800

σ0

σ (MN/m2)

400 Zr – 10% Ti(α′) Zr – 15% Ti(α)

600 500

Zr – 10% Ti(α)

400

Zr – 5% Ti(α′)

Δσ0 (MN/m2)

700

500

σ1 – σ0

Zr – 5% Ti(α)

300

200

300 Zr(α′) 200

Zr(α)

100 0

100

1

ε1 2

Slope 0.12 MN × m 0

0

5

10

20

15

ε

1

2

25

30

0

[(λ 2)

–1

× 102

(a)

1

2

– (λ1)

–3

2 –1

2

] (μm)

2

3 –1

2

(c) 1.8 1.6

E

σ1 – σ0

× 103

1.4 1.2 1.0 0.8 0.6 0.4 0.2

Slope 0.19 0

1

2

3

4

( bε1) λ

½

5

6

7

8

× 103

(b)

Figure 4.40. (a) A plot of versus 1/2 , (b) a plot of 1 − 0 /E vs b/ 1/2 / and (c) a plot 1/2 1/2 between 0 and 2 − 1 .


337

the second segment. In martensitic structures, the initial flow showing a high work hardening rate can be attributed to the localized flow of some regions of high stress concentration, while the second segment corresponds to the parabolic stress–strain curve characteristic of the deformation regime where work hardening is essentially controlled by the short-range interaction between gliding dislocations and the steadily increasing geometrically necessary dislocations. This has been confirmed by demonstrating a linear relation (Figure 4.40(b)) between the two dimensionless quantities (1 − 0 /E and (b/ 1/2 , where 1 is the stress corresponding to a plastic strain 1 , 0 is the extrapolated value of the true stress at = 0 (as shown Figure 4.40(b) and is the mean free slip length between the partitioning interfaces which are impenetrable to dislocations. The parameter corresponds to the average width of Widmanstatten plates for the slow-cooled structure and to the average packet size and twin spacing in the lath and the twinned martensite, respectively. Substituting M = 4, a value proposed by Armstrong et al. (1962) for hcp metals, the theoretical slope of a plot of (1 − 0 /E versus (b1 / 1/2 can be estimated to be 0.17 which is quite close to that experimentally obtained. This analysis points to the fact that the enhanced work hardening in internally twinned Zr–Ti martensites is due to the presence of closely spaced partitioning interfaces (which are predominantly {101¯ 1} twin boundaries). The 0 values, obtained form the extrapolation shown in Figure 4.40, closely match with the 0.2% yield stress values for both the martensitic and the Widmanstatten structures. In order to examine whether the influence of the geometric slip distance, , on the yield stress of these structures follows the Hall– Petch relation, 0 values for a given alloy for the and structures, represented by 0 ( ) and 0 , respectively, can be expressed as follows: & 0 = i + KX −1/2 1

0 = i + KX −1/2 2 0 = 0 − 0 = KX −1/2 − −1/2 2 2

(4.61)

In these equations, i is the friction stress corresponding to zero strain value, KX is the Hall–Petch constant and the geometric slip distances for the and the structures are 1 and 2 , respectively. It is implicit in Eq. 4.59 that the parameter i has nearly the same values for the and the structures for a given alloy composition in spite of the difference in the dislocation densities in these. The Hall–Petch constant KX depends on alloy composition mainly through its dependence on the shear modulus, . Since the shear moduli for Zr and Ti are not much different, KX is not expected to alter significantly with alloy composition in the binary Zr–Ti system. This justifies a linear relation between − −1/2 ), as shown in Figure 4.40(c). The slope of this plot has been 0 and ( −1/2 2 1

338


found to be close to the reported value of the Hall–Petch constant (Ramani and Rodriguez 1970) in Zr alloys at 300 K. The applicability of the Hall–Petch relation for estimating the increase in 0 due to the introduction of the martensitic structure demonstrates that the refinement of the grain size factor essentially controls the martensite strengthening in this system. The flow stress, , can be expressed in terms of several components as per different criteria chosen, viz., thermally activated and athermal, dependent on or independent of grain size, plastic strain and solute content. Banerjee et al. (1978) have examined how the martensitic structure influences the different components of the flow stress of Zr–Ti alloys. On the basis of the preceding analysis, the flow stress, , can be expressed as = 1 + stat + KX 1/2 + Cb/ 1/2

(4.62)

where i is the friction stress, C represents the constant term C1 EM −1/2 (see Eq. 4.48) and stat is the -independent component of the flow stress arising from the statistically stored dislocations accumulated during plastic flow. The contribution of stat in the case of Zr–Ti martensites has been shown to be very small and can be neglected. Stress relaxation experiments have been used for measuring the athermal component, M , of the flow stress, and the thermal component, ∗ , of the flow stress has been obtained by subtracting from the total flow stress, (from the relation, = + ∗ ). The values of the activation area, A∗ , associated with the thermally activated flow in both Widmanstatten and martensitic structures for these alloys have also been determined from the stress relaxation data. The salient features of the results arrived at are as follows: (1) The contribution of the martensitic structure (for both the dislocated lath and the twinned plate martensites) to strengthening is mainly through an increase in the athermal component, , of the flow stress. (2) The thermal component, ∗ , of the flow stress is essentially independent of the Ti concentration in these Zr–Ti alloys, suggesting that the strengthening due to the substitutional solid solution arises mainly from an increase in the long-range athermal stress. This is consistent with the results obtained in a number of Zr- and Ti-based substitutional alloys (Conrad et al. 1972). (3) The value of ∗ is independent of plastic strain, implying that dislocation– dislocation interactions do not constitute the rate-controlling mechanism of thermally activated deformation. (4) The values of the activation area for both the and structures in Zr– Ti alloys remain within a narrow band between 25b2 and 35b2 where b2 is


339

the Burgers vector. This implies that the rate-controlling mechanism of the thermally activated flow remains unaltered with changes in the Ti content of these alloys and with the introduction of the martensite structure. The values of the activation area in the case of these martensites closely match with that reported for annealed Zr containing about the same levels of interstitials (about 600 ppm O2 and 60 ppm N2 ). Based on the results of stress relaxation experiments, it could be inferred that the rate-controlling thermally activated deformation mechanism is not altered by the introduction of the martensitic structure and even by the addition of substitutional alloying elements. This is consistent with the findings of Conrad et al. (1972) who have demonstrated that substitutional alloying elements and / phase distribution do not affect the rate-controlling deformation mechanism in Ti-based alloys. They have suggested that a mechanism involving dislocations overcoming the interstitial solute obstacles is rate controlling during low temperature deformation of Ti alloys. However, there is some controversy over this issue, and an alternative view is that the Peierls–Nabarro barrier, which is increased by interstitial additions, is the rate-controlling obstacle. The fact that ∗ is not affected by the introduction of the martensitic structure and the extent of plastic strain indicates that ∗ is contained within the friction stress i (Eq. 4.59), which can be resolved into two components, ∗ and j (X). The former is essentially controlled by the interstitial level while the latter, a part of the athermal stress, , is independent of and and depends on the nature and the extent, X, of substitutional alloying additions. The total flow stress can, therefore, be expressed as = ∗ + j X + KX −1/2 + Cb/ 1/2

(4.63)

While the increments in the initial yield stress and the work hardening rate due to the introduction of the martensite structure are given by the terms KX −1/2 and Cb/ 1/2 , respectively, the increments in the initial yield strength due to interstitial and substitutional solid solution hardening are reflected in the terms, ∗ and j (X), respectively.

4.5

MARTENSITIC TRANSFORMATION IN Ti–Ni SHAPE MEMORY ALLOYS

Near-equiatomic Ti–Ni alloys and many of their multicomponent derivatives are well known for exhibiting the shape memory effect and pseudoelasticity. In these

340


Ti–Ni-based alloys, three martensitic phases occur. The fully annealed binary Ti– Ni alloys transform from the B2 parent phase directly to the monoclinic B19 phase. In contrast, thermomechanically treated Ti–Ni alloys transform in two steps, namely the B2 phase to the trigonal R-phase (Goo and Sinclair 1985) and the R-phase to the monoclinic B19 phase (Wang et al. 1968). Ternary Ti–Ni alloys containing a few percentage of Fe or Al undergo the same sequence, B2 → R → B19 , of transformations (Mitsumoto and Honma 1976). Ti-Ni-Cu alloys, in which Cu substitutes Ni by 5–15 at.%, also transform in two steps; but in this case, the B2 phase first transforms into an orthorhombic B19 phase which finally transforms into the monoclinic B19 phase (Nam et al. 1990). The basic crystallography associated with these transformations has been discussed in the first part of this section. Some of the important microstructural characteristics of these martensites, which are essentially responsible for imparting the shape memory property, are also highlighted. The microstructural feature which is common to all shape memory alloys is such that the martensite plates are spatially organized in a very effective (Saburi and Nenno 1982) accommodating manner. The issue of self-accommodation in the martensite microstructure has been introduced earlier in the context of martensite morphology. The importance of both macroscopic self-accommodation and microscopic self-accommodation in the martensitic microstructure of Ni–Ti alloys, as reported recently by Madangopal (1997), has also been discussed in this section. The shape memory effect is intimately linked with some associated phenomena such as thermoelastic equilibrium, pseudoelasticity and thermal arrest memory effect. An account of these related phenomena is included in this section mainly to explain the interrelationship of these in terms of the microstructural response of shape memory alloys to alterations in the applied stress and temperature. 4.5.1 Transformation sequences The transformation temperatures and the sequence of transformation of Ni–Ti shape memory alloys are known to be sensitive to the chemical composition of the alloy, the memory imparting heat treatment, the prior cold work and the externally applied stress. Differential scanning calorimetry (DSC) and resistivity measurements have been mainly used for detecting the transformation sequences in Ni–Ti alloys with varying compositions and prior thermomechanical histories. A fully annealed, near-equiatomic NiTi alloy shows only a single peak in the DSC thermogram (Figure 4.41) during either a continuous heating or a continuous cooling experiment, indicating that both the forward B2 → B19 and the reverse B19 → B2 transformations occur in a single step. The same material, however, after being cold worked to a level of about 15% reduction in thickness, exhibits a two-step transformation during cooling, the first and the exothermic peaks representing the B2 → R and the R → B19 transformations. The transformation during heating


R/R293 K

(a)

341

Ms

1.4

Af

Mf 1.0

TR As

(b)

Heat flow Endothermic exothermic

1.6 × 10–3 J/s

Af Heating

Cooling TR

Ms 170

220

270

320

370

Temperature, T (K)

Figure 4.41. DSC thermogram showing transformation temperatures and the sequence of transformations in a Ni–Ti shape memory alloy.

still occurs in a single step, giving rise to a single endothermic peak. The temperature gap between the B2 → R and the R → B19 transformations increases with an increase in the extent of the prior cold work. The appearance of the R phase during the cooling of cold-worked NiTi alloys can be detected by monitoring electrical resistivity as a function of temperature or during in situ cooling of a sample in a TEM. Electron diffraction patterns, obtained from samples comprising the B2 phase at a temperature close to the B2 → R transition, show diffuse intensity with maxima located at reciprocal lattice positions which divide the B2 reciprocal lattice vectors in three equal segments. The intensity maxima at the 1/3 positions remain very weak and dark field micrographs using such reflections fail to reveal any observable contrast. However, as the sample is cooled further, plates of the R

342


phase appear progressively and these plates get organized in a self-accommodating configuration which is described later. In situ experiments have also indicated that R-phase plates can nucleate at relatively small strain centres, such as single dislocations. As these plates grow and gradually fill up the B2 grain, sharp 1/3 reflections appear in the diffraction patterns. These reflections can be indexed on the basis of the trigonal crystal structure of the R phase. On further cooling, the R phase transforms into the B19 phase. Crystallographic features of these two martensitic transformations are discussed in the next section. 4.5.2 Crystallography of the B2 → R transformation The lattice correspondence between the B2 and the R phases is given in Table 4.12 which lists the relative dispositions of the four crystallographic variants of the R phase with respect to the parent B2 phase. Self-accommodation is achieved in this transformation by an assembly of the variants which is illustrated in Figure 4.42. While between variants 1 and 2 and between variants 3 and 4 there ¯ R type compound twinning, that between variants 2 and exists a relation of {1122} 3 and between variants 1 and 4 is of {1121}R type compound twinning. As shown ¯ R -type twin planes are derived from {100}B2 type in Figure 4.42(b), the {1121} mirror planes and the {1122}R type twin planes are derived from {110}B2 -type mirror planes of the parent phase. It is also evident from Figure 4.42 that six kinds of equivalent arrangements are possible (two for each of the axes of the parent phase) among the four variants generated from a single parent crystal. Two distinct morphologies, as illustrated in the light micrograph in Figure 4.42, are possible, one characterized by a stack of twin-related bands and the other by a herring bone morphology in which the four variants meet along an invariant line parallel B2 axis. In both cases, the planar interfaces separating adjacent variants are all twin planes. 4.5.3 Crystallography of the B2 → B19 transformation The lattice correspondence between the B2 structure and the orthorhombic B19 structure is shown in Figure 4.43, in which i , j and k represent the orthorhombic Table 4.12. Lattice correspondence between B2 (parent) and R phases. Variant

[100]R

[010]R

[001]R

(001)R

1 2 3 4

¯ p [121] [211]p [121]p [211]p

¯ p [112] ¯ ¯ p 112 [112]p ¯ p [112

[111]p ¯ p 111 ¯ p [1¯ 11] ¯ p [111]

[111]p ¯ p [111] ¯ p [1¯ 11] ¯ p [111]


343

1 4 1 4 1 1 2 1 2 1 2 1 2 1

4 4 1 2 23 4 1 42 23 4 4 1 2 2 3 (a) [001] [010]

[100] (110) (010)

(110)

4 3

2 4 (b)

Figure 4.42. (a) Schematic of an optical micrograph of self-accommodation of the R phase. The variant numbers are assigned on the basis of the lattice correspondence described. (b) Threedimensional arrangement of self-accommodating R-phase variants.

axes while i, j and k correspond to the cubic axes. The k -axis can be made parallel to one of the cube axes, and in such a case, the i and j axes can be oriented in two distinct manners. Therefore, six possible correspondence variants are present. The relative orientations of these six variants, numbered 1–6 with respect to the axes system of the parent B2 crystal, are indicated in Table 4.13. The homogeneous deformation matrix, T , which is associated with the B2 → B19 transformation is given by ⎤ ⎡ a 0 0 √ ⎥ ⎢ T = 1/ao ⎣ 0 b/ 2 0 ⎦ (4.64) √ 0 0 c/ 2

344

Phase Transformations: Titanium and Zirconium Alloys Ni atoms k,k ′

Ti atoms

j

i′

j′ i′

aj ′ k,k ′

β

bj′ cj′

Figure 4.43. The lattice correspondence between the B2 structure and the orthorhombic B19 structure.

Table 4.13. B2-martensite lattice correspondence. Variant

[100]m

[010]m

[001]m

1 1 2 2 3 3 4 4 5 5 6 6

[100]c ¯ c 100 [100]c ¯ c 100 [010]c ¯ c 010 [010]c ¯ c 010 [001]c ¯ c 001 [001]c ¯ c 001

[011]c ¯ c 01¯ 1 ¯ c 011 ¯ c 011 ¯ 101c ¯ c 101 [101]c ¯ 1 ¯ c 10 ¯ c 110 ¯ c 110 [110]c ¯ c 1¯ 10

¯ c 011 ¯ c 011 ¯ c 01¯ 1 ¯ ¯ c 011 [101]c [101]c ¯ c 101 ¯ c 101 ¯ c 101 [110]c ¯ c 110 ¯ c 110


345 (110)P

(100)P

1

6

(010)P

3 4

2 5

[001]

[100]

(101)P

2

4 6

[010]

(a)

(001)P

(011)P

(b)

Figure 4.44. Self-accommodation model of B19 phases.

where ao is the lattice parameter of the B2 structure while a, b and c are the lattice parameters of the B19 structure. Two types of twin relations, namely (111) [211]O type I and (011) [01¯ 1]O compound twins, exist between variants which come in contact along a planar surface. The subscript O refers to the orthorhombic B19 structure. Four 111O -type twin interfaces (derived from parent 110B2 planes) and six 011O -type twin interfaces (derived from 100B2 planes) separate adjacent variants in the self-accommodating assembly of martensite variants produced in this transformation. Self-accommodation model of B19 phases is shown in Figure 4.44. 4.5.4 Crystallography of the B2 → B19 transformation The B2 → B19 transformation can be conceptually divided into two components, namely the B2→ orthorhombic B19 and the B19→ monoclinic B19 transformations. As discussed in the previous section, six possible variants of lattice correspondence are possible between the B2 and B19 structures. The lattice shear which transforms the orthorhombic structure into the monoclinic structure is a simple shear acting on the (100)O plane along the [001]O direction. Depending on ¯ O direction, the sign of this shear, i.e. whether it is along the [001]O or the 001 two monoclinic variants can be generated from each of the orthorhombic variants; these monoclinic variants are designated as 1,1 , 2,2 , 3,3 , 4,4 , 5,5 and 6,6 . Therefore, a total of 12 variants can form from a single B2 crystal and their relative orientations are also indicated in Table 4.13. The homogeneous deformation matrix, T , which converts the parent B2 crystal into the monoclinic martensite is given by √ ⎤ a 0 c cos / 2 √ ⎥ ⎢ 0 T = ao ⎣ 0 b/ 2 ⎦ √ 0 0 c sin / 2 ⎡

(4.65)

346


where defines the angle between the a- and the c-axes. Considering the case of a specific variant, 6 , the Bain deformation B6 can be calculated by incorporating the necessary axis transformation matrix R6 : ⎡ ⎤ 0 1 1 R6 = ⎣ 0 −1 −1 ⎦ (4.66) −1 0 0 B6 can be expressed as B6 = R6 T R6 −1 and can be shown to be ⎤ ⎡ i j 0 ⎥ ⎢ (4.67) B6 = ⎣ j i 0 ⎦ l l k √ √ where j = 2/4 amo −b + c sin , k = c cos /2 amo , l = a/ao and i = 2/4 ao b + c sin Substituting the following values of lattice parameters (Miyazaki et al. 1983), ao = 03015 nm, a = 02889 nm, b = 04120 nm, c = 04662 nm, and = B6 can be evaluated as ⎤ ⎡ 10213 00550 00 ⎥ ⎢ (4.68) B6 = ⎣ 00550 10213 00 ⎦ ‘00907 ‘00907 09582 Bain deformations for the other variants can be computed using the appropriate axis transformation matrices. The LIS in these martensites invariably occurs by twinning. As will be discussed later, the propagation of twin interfaces has a pivotal role to play in the process of shape recovery. The basis of the selection of the twinning systems and their experimental identification are briefly covered here. In order to satisfy the condition of equivalent lattice correspondences for the two twin components of a martensite plate, it is necessary that either the twin plane (K1 plane) is derived from a mirror plane of the parent crystal or the 2 direction is derived from a diad of the parent. Examining all the possible twin systems, Madangopal and Banerjee (1992) have arrived at the conclusion that only ¯ M type I, the conjugate [112]M type II, (011)M type four twin systems, viz., 111 I and the conjugate [011]M type II can qualify to be the lattice invariant shear system in this case. Examples of type I and type II twins in the Ni–Ti martensite are illustrated in Figure 4.45. Since type I and type II twins are conjugate to each other, the twinning shear is the same in the two twinning modes. In type I twinning


347

Figure 4.45. TEM micrographs showing the (a) type I and (b) type II twinning as the lattice invariant shear.

the K1 plane, being of rational indices, will be fully coherent. In contrast, the irrational K1 planes of type II twins generally consist of rational ledges and steps (Figure 4.45). In this respect, type I twin interfaces are energetically favourable, and as a consequence, the mean spacing between the twin interfaces is usually much smaller for type I twins than that for type II twins. Early TEM observations on the as-quenched Ni–Ti monoclinic martensites have ¯ M type I and/or {001}M compound twins (Otsuka shown the presence of 111 et al. 1971, Gupta and Johnson 1973). Knowles and Smith (1981a,b) have detected Md , the stress–strain plot shows elastic deformation up to a high stress value, followed by a limited plastic flow and finally a sudden brittle fracture. The parent B2 phase is stable with respect to any phase transformation at T1 , and thus the observed deformation behaviour is consistent with that expected from a brittle ordered intermetallic compound. At the testing temperature, T2 Md > T2 > Af , the B2 phase is unstable with respect to the stress-induced martensitic transformation which is induced when the stress attains a threshold value indicated by the point 1 on the stress–strain plot. The plastic flow from 1 to 2 corresponds to the formation of increasing volume fractions of the martensite. While unloading the


353

M f > T3 T1 > Md

Md > T2 > Af Pseudoplastic deformation

8 8′

2 1

Stress

3 4

6

5

Recoverable pseudoplastic strain

d a

c

7 10

9

b Str

ain

Mf M s ure t a r pe em

s Af A

T

Figure 4.49. A schematic drawing showing shape memory effect and pseudoelastic deformation in alloys exhibiting martensitic transformations.

sample, the stress falls from 2 to 3 in a manner similar to elastic unloading. At the point 3, the volume fraction of the stress-induced martensitic starts decreasing and the stress–strain plot follows the path indicated by 3 → 4. The closed stress–strain hysteresis loop indicates that the stress-induced martensitic structure completely reverts back to the parent phase during unloading. The non-linear deformation which is recoverable on unloading is known as pseudoelastic deformation. The stress–strain plot at T3 T3 < Mf shows a deviation from elastic deformation at a fairly low stress value, resulting in a deformation plateau (indicated by 5 → 6). Unloading from the point 6 shows an elastic unloading to the point 7. The plastic strain at the point 7, however, is recoverable by heating the sample to a temperature higher than Af and hence is known as pseudoplastic strain. The recovery of the pseudoplastic strain starts on heating to a temperature above Af . This strain recovery (shape recovery) process of a pseudoplastically deformed material, when subjected to a heating cycle to go through the parent phase, is known as the shape memory effect. In case deformation at T3 is continued beyond the point 6, a second stage of linear elastic deformation is encountered. The stress rises substantially to reach the point 8 where a deviation from linearity is noticed. Specimens fracture at 9

354


after plastic flow occurs to a limited extent. Unloading from a point such as 8 results in elastic recovery of strain in a linear manner (8 to 9). On subsequent heating above Af , the strain recovery occurs only to a limited extent (9 to 10). Components of some shape memory alloys, after being subjected to a number of straining and thermal cycles (for strain recovery), acquire a two-way memory. In such cases, the component (or the specimen) remains in two states of strain (or two shapes) at two temperatures, one above Af and the other below Mf . The two-way shape memory is also denoted by a strain – temperature cycle, abcd, as shown in Figure 4.49. The schematic drawing (Figure 4.49) shows in an idealized manner different phenomena, namely pseudoelastic deformation (or rubber-like behaviour), pseudoplastic deformation, shape memory and two-way memory effects. Intensive research on these phenomena during the last four decades has established the sequence of structural changes which occur during these processes and are described in detail in several reviews (Delaey et al. 1974, Schetky 1979, Wayman 1980, Tadaki et al. 1988). Physical processes accompanying these individual steps are not exactly identical for all the shape memory alloys. In spite of the differences between the systems, one can attempt an oversimplified description of the processes responsible for the aforementioned phenomena. Such an attempt is made in the following paragraphs. At a temperature between Af and Md , the latter being the upper temperature limit for the stress-assisted martensitic transformation, the austenite to martensite transformation can be induced when the chemical driving force for the transformation is augmented by the applied mechanical stress. The deviation from the linear elastic behaviour is noticed at a value of stress which is adequate for initiating the stress-assisted transformation. With a further increase in stress, the volume fraction of the martensite phase increases, the mechanical work done on the system by the applied stress being entirely used up for the creation of the metastable martensitic phase. The martensitic phase forming under this condition remains in thermoelastic equilibrium, and it is, therefore, possible to reverse the process (i.e. to induce the martensite to austenite transformation) once the stress level is lowered. The unloading path, as shown by 2-3-4, consists of pure elastic unloading from 2 to 3 followed by the martensite to austenite reversion from 3 to 4. The loop, therefore, represents the hysteresis in the stress–strain relation with respect to the stress-assisted martensitic transformation. Deformation of the fully martensitic structure, at temperatures below Mf , shows pseudoplastic flow (from point 5 to 6) which can be completely recovered by the thermal cycle as described earlier. This can be contrasted with the usual plastic flow of metals and alloys by slip in which the slipped regions of crystalline grains are shifted to new positions having identical atomic surroundings. Since the slip process


355

is not reversible, there is no tendency of the slipped regions to retrace the deformation path. The slip mechanism cannot, therefore, explain the recoverable pseudoplastic deformation. The starting structure, being fully martensitic, consists of an aggregate of different variants of plates which are self-assembled in a manner such that the strain energy of the assembly is minimized. It has also been shown that a majority of the intervariant interfaces satisfy a twinning relationship across them. On application of an external stress, some of the orientation variants grow at the expense of their neighbouring orientations. Under the given system of the external stress, some variants are favourably oriented for growth while others are not. Though pseudoplastic flow can be accomplished by such a reorientation process which does not revert during unloading, the self-accommodation achieved in the virgin martensitic structure is disturbed. A thermal cycle through the austenite phase brings back the self-assembly and in that process nullifies the pseudoplastic strain. A component of a shape memory alloy, when cycled a number of times through the process of pseudoplastic deformation and shape recovery, develops a twoway memory. Such a component assumes two shapes given by two states of strain corresponding to two temperatures, one below Mf and the other above Af . The two-way memory arises due to accumulation of residual plastic deformation in the material during the course of repeated thermal cycling. The locked-in residual stress eventually builds up to such a level that it can drive pseudoplastic deformation without the introduction of any externally applied stress. The shape recovery occurs in the usual manner during the heating-up step. In the foregoing paragraphs, it has been brought out that the martensitic transformation plays the pivotal role in bringing about shape memory and related phenomena. It is also true that all systems, metallic or ceramic, in which martensitic transformations occur do not exhibit the shape memory effect with a significant extent of shape recovery. It is, therefore, important to identify the criteria which must be fulfilled for a system to display shape memory up to a few percentage of pseudoplastic strain. These criteria are as follows. (a) The first and foremost condition is that the system must display a thermoelastic martensitic transformation. This condition essentially implies that the magnitude of transformation strain is not large enough to cause irreversible plastic flow in either the matrix or the martensitic phase. (b) Martensite plates in the product microstructure must form a self-accommodating assembly. The minimization of the overall strain energy results in a strain coupling between different martensite plates. (c) Interfaces between adjacent martensite plates should be such that they can propagate either way without losing atomic registry. Such a condition is met if a great majority of these interfaces satisfy a twin relationship across these interfaces. (d) Atomic ordering of both the austenite and the martensite phases favours the occurrence of the shape memory effect, though this is not an essential requirement. The presence

356


of atomic ordering restricts the number of variants for the martensitic transformation and also increases the threshold stress at which thermoelastic reversibility is lost due to plastic deformation. The criteria which are listed here are based on Ni–Ti, In–Tl, Au–Cd and Cubased shape memory alloys, which were the systems in which this phenomenon was reported initially. In recent times, this phenomenon has been observed in several other alloys such as intermetallics and ceramic systems. A number of Fe-based alloys including Fe-Ni-C, Fe-Mn-Si, Fe-Mn-Si-Cr-Ni and Fe-Ni-Co-Ti have shown shape memory effect to somewhat restricted recoverable plastic strain values. Currently, the search for shape memory alloys with higher recovery temperatures and larger hysteresis is being pursued in several laboratories. Intermetallics based on Ni–Al and Ti–Pd show good promise. 4.5.7 Reversion stress in a shape memory alloy The performance of a shape memory device depends not only on the maximum limit of the recoverable strain but also on the stress against which the strain recovery can occur. The reversion stress r is defined as the stress developed in a sample constrained against recovery during heating to a temperature higher than Af . Data available on reversion stress and on its dependence on prior plastic strain and temperature have been very limited. The method of the measurement of the reversion stress in shape memory alloys and the criteria which determine the upper limit or r have been reported by Madangopal et al. (1988). The experiments involved (a) deforming fully martensitic tensile test samples of Ni–Ti shape memory alloys to a plastic strain p using a hard tensile testing machine at a temperature below Mf , (b) unloading the sample to just above the zero load level, (c) arresting the crosshead of the testing machine at this position, (d) rapid heating of the sample to a temperature Tr Tr > Af by introduction of a hot water bath and recording the rise in stress as the sample underwent the reversion process and (e) unloading from the maximum stress level, r , maintaining the sample temperature at Tr . These steps are schematically illustrated in Figure 4.50(a). Thermal stresses introduced due to differential thermal expansion of the sample and the loading assembly have been determined to be 15 MPa, which is much smaller compared to the measured r values (100–500 MPa). The influence of p and Tr on r and its saturation value is shown in Figure 4.50(b). Figure 4.51(a) shows r values attained for different levels of initial plastic strain and the unloading paths (pseudoelastic) after attainment of r . The shape of this unloading curve closely matches with the pseudoelastic unloading curve (outer envelope of the stress–strain hysteresis loop shown in Figure 4.51(a). This suggests that the thermoelastic equilibrium attained after the development of r in a constrained recovery process is similar to (close to but not identical


M d > T2 > A f

Stress (σ) →

σr

357

T1 < Mf

Stress (ε) → (a)

Reversion stress (MPa), σr→

600

Equiatomic Ni Ti SMA As ~ 323 K Ms ~ 318 K

353 K 369 K 378 K 398 K 0 423 K

00 0

500

0 0 400

0 300

0 200

100

1

2

3

4

5

6

Constraint (%), εp →

7

8

(b)

Figure 4.50. (a) Schematic stress–strain plot showing the different steps involved in the measurement of the reversion stress. (b) The variation of the reversion stress with temperature and prestrain.

358

Phase Transformations: Titanium and Zirconium Alloys Tr ~ 313 K

Stress (MPa) →

350 300 250 200 150 100 50 0

0

0.5

1

1.5

2

2.5

3

Strain (%) → (a) 322 K

300

σr →

Stress (MPa) →

318 K

200

313 K 308 K 303 K

100

– Crosshead Stationary only chart movement – Thermal perturbations 10

Strain (%) → (b)

Figure 4.51. (a) The unloading path from superimposed on the correspondence pseudoelastic loading–unloading loop. (b) The change in with thermal perturbation.

with) that prevailing in the unloading path of the pseudoelastic stress–strain loop. Therefore, it is surmised that r is nothing but the stress threshold which marks the start of the reversion of stress-induced martensite (M) into austenite (B2). The fact that a thermoelastic equilibrium is reached after the attainment of r is elegantly demonstrated in an experiment in which growth of martensite phase and a consequent drop in stress have been recorded as the sample temperature is slightly lowered by spraying cold water on the sample (Figure 4.51(b)). The important observations listed above point to the fact that on the attainment of r , the M → B2 transformation gets arrested and a thermoelastic equilibrium is established. The stress value at which the stress-induced martensite starts reverting back to the parent austenite phase is generally designated as As which is marked

Martensitic Transformations 500

Tr > As

σMs

Stress (MPa), σ Ms,σAs,σr →

400

359

σ

σr σMs

σAs

σAs

ε 300

200

100

Ms 0

As

283 287

Af 300

310

320

330

Temperature (K) Tr →

Figure 4.52. The variation of As , Ms and r∗ with temperature.

on the pseudoelastic unloading path in Figure 4.52. This stress value essentially corresponds to the stress at which the reverse M → B2 transformation commences during the unloading process. On the other hand, as the temperature is raised above Af in the constrained condition, pseudoplastically deformed martensite progressively transforms to the austenite (B2) phase and the elastic stress developing in the system (consisting of a mixture of M and B2 phases) gradually rises. This reversion stress, however, cannot rise above that corresponding to the M → B2 thermoelastic equilibrium. This explains why r attains a maximum value of r∗ which equals As . It is possible to predict r∗ by using a modified Clausius–Clapeyron equation relating As and the reversion temperature, Tr : dr∗ Hp =− dTr T

(4.69)

where H is the change in enthalpy for the B2→ M transformation, is the maximum pseudoplastic strain, T is the Af temperature and p is the density of the B2 phase.

360


4.5.8 Thermal arrest memory effect A new phenomenon named thermal arrest memory effect (TAME) reported in Ni–Ti shape memory alloys is briefly discussed here. When the martensite to parent (M → P) transformation is arrested for a short duration at a temperature, TA , before completion of the transformation (i.e. As < TA < Af and cooled down to a temperature below Mf , the martensite to parent (M → P) transformation is halted during subsequent heating at the same temperature, TA . This means that the system remembers the temperature at which the transformation was initially arrested in the preceding heat treatment cycle. The phenomenon can be seen clearly in continuous heating and cooling experiments in a differential scanning calorimeter (DSC). DSC thermograms illustrating the exothermic (P→M) and the endothermic (M→P) transformations are shown in Figure 4.53(a). The steps involved in the experiment are described below. (Arrest in the martensite to parent M→P transformation) Sequence A (Figure 4.53(b)) Step 1: The M→P transformation is arrested at a temperature

NiTi Thermal cycle:

1

ΔH, heat flow rate (arbitrary units)

+

2

260 K

3

A Thermal arrest during P-M

4 360 K

M→ P

Thermal arrest

–

B during M-P

M← P B A

–

(a)

280 K 1

2

+

+

360 K

Thermal arrest 337.1 K

1

2

Thermal arrest

4

275 K

360 K

(b) 3 360 K

280 K

– 3

4 (c)

Temperature

Figure 4.53. DSC thermogram exhibiting the effect of interrupting the M-P and P-M transformations in Ni50 Ti50 alloy.


361

TA = 3371 K, between As and Af . Step 2: On cooling from 337.1 K to a temperature below Mf , the parent phase generated in step 1 is again transformed to martensite. Step 3: During heating in the next cycle, the M→P endotherm splits into two, centered at TA . Step 4: During cooling from a temperature above Af , the original exothermic peak for the P→M transformation is obtained. (Arrest of the parent to martensitic (P→M) transformation) Sequence B [Figure 4.53(c))] Step 1: The P → M transformation is interrupted at TA (= 314.8 K) between Ms and Mf . Step 2: On heating to a temperature above Af , the partially transformed martensitic microstructure reverts back to the parent phase. Step 3: On subsequent cooling to a temperature below Mf , the original exothermic peak is obtained without showing any effect of the thermal arrest in step 1. Step 4: During heating the original exothermic peak for the complete M→P transformation is observed. These observations clearly demonstrate that a thermal arrest during the M→P transformation is remembered by the system during the next heating cycle. Such is not the case for a thermal arrest during the P→M transformation. However, in cold-worked samples, TAME is observed for arrest of both P→M and M→P transformations. A detailed study of the influence of various factors like supercooling, premartensitic R-phase transformation, repeated thermal arrest of TAME has been reported by Madangopal et al. (1994). Taking into account all these observations, the thermal arrest memory effect has been rationalized as follows: The split in the M→P transformation, as seen in Figure 4.53(b), step 3, implies that the M→P transformation stops for a short duration at the arrest temperature TA in the previous cycle. As discussed earlier, martensite plates tend to minimize the elastic strain energy of the system by self-accommodation. Plates belonging to different generations (forming at different stages of the transformation) are elastically coupled. An interruption in the transformation, therefore, results in dividing the population of martensite plates into two distinct groups. At TA , the temperature of arrest, the system consists of a mixture of the martensite and parent phases. On subsequent cooling, martensite plates transforming from this austenitic region form a distinct (second) group. The M→P transformation for these two groups of martensite does not remain linked during the second cycle. The reverse transformation of the second population occurs first and only after the completion of the transformation of this group, the M→P transformation of the other group commences. This sequence results in the introduction of a break in the endotherm corresponding to the M→P transformation. The thermal arrest memory effect is directly linked with the self-accommodation process which binds one generation of martensite plates with the next by elastic coupling. In case the self-accommodation in a small group of three or four plates is very effective, the magnitude of the unaccommodated strain will be small,

362


and consequently the extent of elastic coupling between generations of martensite plates will also be small. Such a situation prevails in Cu-Zn-Al martensites where self-accommodation of neighbouring martensite plates in a group is as high as 98%. TAME is, therefore, observed rather weakly in this system.

4.6

TETRAGONAL MONOCLINIC TRANSFORMATION IN ZIRCONIA

4.6.1 Transformation characteristics Polymorphic transformations in pure zirconia (ZrO2 ) have been briefly described in Chapter 1. This transformation, first detected by Ruff and Ebert (1929) using high-temperature X-ray diffraction, has been extensively studied using thermal analysis, X-ray and electron diffraction, light and electron microscopy and electrical resistivity measurements. Wolten (1963) was the first to suggest that the monoclinic to tetragonal transformation is martensitic in nature. The important characteristics of this transformation are summarized in the following. (1) The high-temperature tetragonal phase cannot be retained on quenching to room temperature. (2) The transformation is athermal. (3) The observed growth rate of monoclinic platelets has been found to be consistent with a velocity of the transformation front approaching that of sound in solids (Fehrenbacher and Jacobson 1965). The transformation kinetics, as measured from thermal analysis experiments, show a burst-like behaviour (Maiti et al. 1972) during the reverse transformation (i.e. monoclinic to tetragonal). (4) The transformation exhibits a large thermal hysteresis (Figure 4.54); the forward and the backward transitions occurring during cooling and heating experiments show transition temperatures in the vicinity of 1123 and 1323 K, respectively (Figure 4.54). (5) The monoclinic to tetragonal transformation is accompanied by the sudden appearance of surface relief which corresponds to the formation of tetragonal plates during heating experiments. (6) A strict orientation relationship exists between the parent and the product phases, as described in detail in the next section. The product phase, either plate-shaped or lenticular-shaped, always forms along specific habit planes. The lattice correspondence derived from the observed orientation relationship suggests that the transformation can be accomplished by atomic movements, primarily of O atoms, to an extent smaller than the interatomic distance and minor shifts of Zr atoms.


363

100 Decrease temperature

Percent tetragonal phase

80 Increase temperature 60

40

20

0

600

700

800

900

1000

1100

1200

Temperature (°C)

Figure 4.54. Monoclinic to tetragonal transformation exhibiting a large thermal hysteresis during cooling and heating.

All these experimental observations strongly point to the fact that both the forward and the backward transformations can occur by a martensitic mode when the cooling or the heating rate exceeds a certain critical value. 4.6.2 Orientation relation and lattice correspondence There has been some initial disagreement in literature on the exact orientation relation between the monoclinic and the tetragonal phases (Bailey 1964, Wolten 1964, Smith and Newkirk 1965, Patil and Subbarao, 1967). The experimental difficulties due to the high temperature of the transformation have been mainly responsible for this discrepancy among results initially reported by these different investigators (Bansal and Heuer 1972). They devised an ingenious experiment in which single crystals of monoclinic ZrO2 were grown from a fluxed melt at temperatures below the tetragonal to monoclinic transformation temperature. These single crystal monoclinic samples were heated to temperatures above the As temperature for the monoclinic to tetragonal transformation and were subsequently cooled down to a temperature between the Ms and Mf temperatures. X-ray diffraction patterns were recorded in the initial monoclinic state, after the first reverse transformation and after cooling down below Ms . A comparison of the X-ray diffraction patterns obtained from the sample in these three stages yielded the orientation relation between the two phases. These orientation relations are consistent with the possible

364


(a)

LC A

at

at

bm at

LC B

ct

cm am

at

bm

am

at

LC C

at

cm

at

bm

am cm

at (b)

Figure 4.55. (a) Tetragonal and monoclinic structures and (b) three possible lattice correspondences consistent with the orientation relation of the tetragonal and the monoclinic structures of ZrO2 .

lattice correspondences which can be derived from an inspection of the tetragonal and the monoclinic structures Figure 4.55(a) of ZrO2 . Three distinct lattice correspondences can be envisaged as illustrated in Figure 4.55(b). If a right-handed screw convention is adopted for the unit cell axes, three lattice correspondences, designated as types A, B and C, can be defined, depending on which monoclinic axis, am , bm or cm , is parallel to ct , the c-axis of the tetragonal crystal (suffixes “m” and “t” referring to monoclinic and tetragonal structures). The type A lattice correspondence gives rise to the orientation relationship: (001)m

100t and [100]m

< 100 >t ; which has been experimentally observed only for the first reverse (monoclinic → tetragonal) transformation (Bansal and Heuer 1974). For the type A lattice correspondence, no surface relief has been


365

noticed, suggesting that this correspondence is not operative in the case of martensites forming on the surface of monocrystalline samples. The type B lattice correspondence leads to the following two orientation relationships, designated as B-1 and B-2: B-1 * 100m

010t & 010m

001t & 001m 9 to 100t ie 001m

100t B-2 * 100m 9 to 010t & 010m

001t & 001m

100t ie 100m

010t These orientation relations are observed when the crystal is heated to the fully tetragonal phase field and subsequently cooled down to 1273 K (1000 C) where it is partially transformed into the monoclinic phase. The type C lattice correspondence results in the following two orientation relations, C-1 and C-2: C-1 * 100m

100t & 010m

010t & 001m 9 to 001t ie 001m

001t C-2 * 100m 9 to 100t & 010m

010t & 001m

001t ie 100m

100t These orientation relations have been encountered in monoclinic martensite plates forming in the later stages of the transformation (during cooling down from 1273 K 1000 C to room temperature). The directions and magnitudes of the principal distortions associated with the three possible lattice correspondences between tetragonal and monoclinic zirconia can be determined by analytical geometry, as illustrated in Figure 4.56. The principal distortions, 1 , 2 and 3 , are defined along the orthonormal axes parallel to the [100]t , [010]t and [001]t directions, respectively. The problem is two dimensional as one principal axis is obtained directly by inspection (Figure 4.55(b), 3

bm ). The other two principal distortions lie in the plane normal to the 3 direction. The method of determination of principal distortions shows the changes in the relevant tetragonal plane. The

Ym sin β

Yt

Ym

Y Xt X Tetragonal plane

Xm

Xm

(a) Expansion

(b) Shear monoclinc plane

Figure 4.56. Schematic showing various steps of martensitic transformation.

366


Table 4.17. Computed principal strains for three lattice correspondences (LC). LC A

LC B

LC C

1

Direction Magnitude

0 08267 05627 1,0956

0 07383 −06745 0.9337

07860 −06183 0 0.9428

2

Direction Magnitude

0 −05627 08267 0.9287

0 06745 07383 1.0897

06183 07860 0 1.1045

3

Direction Magnitude

100 1.0128

100 1.0128

001 0.9896

distortion of this plane can be factorized into (a) a change in dimensions (expansion/contraction) and (b) a change in shape (simple shear). The calculated values for the principal strains for the three lattice correspondences are given in Table 4.17 (lattice parameters and principal strains). 4.6.3 Habit plane Bansal and Heuer (1974) and Kriven et al. (1981) have analysed the crystallography of the tetragonal to monoclinic transformation in zirconia for the lattice correspondences designated as type B and type C. The systems of lattice invariant shear considered for this analysis are the following: For the type B lattice correspondence: 1¯I0t 001t 110t 1¯I0t 100t 001t 100t 011t For the type C lattice correspondence: 1¯I0t 001t 1¯I0t 110t 10¯It 010t 111t 1¯I0t These shear systems, expressed in terms of tetragonal indices, correspond to slip and twin shears of the monoclinic system. Type A lattice correspondence, being the least likely to be operative in view of the highest lattice strains involved, has not been considered here. Experimental observations support the existence of the correspondences B and C. There is some evidence to indicate that the correspondence B is favoured in the transformation in large grains of the tetragonal phase (Hugo et al. 1988). On the other hand, when the transformation occurs in small particles or fibres, correspondence C is almost always observed (Muddle and Hanmik 1986). The application of the Bowles–Mackenzie (1954) theory with regard to the correspondences B and C yields solutions for habit plane indices, directions and magnitudes of shape strains, a detailed account of which has been presented by Kriven et al. (1981) (Table 4.18).


367

Table 4.18. Predicted habit planes and shape strains. Lattice invariant shear system (010 [101]

¯ (011) [011]

¯ (101)[101]

¯ ¯ (111) [011]

¯ (010 [100] Twin system

Habit plane indices (PI)

Direction of shape strain D1

Magnitude of shape strain M1

−0615 −0334 0.730 ¯ ∼(2¯ 12) −0103 −0988 0.122 ¯ ∼(1¯ 92)

−0067 −0984 0.160 ¯ ∼[06¯ 1] −0620 −0281 0.716 ¯ ∼[7¯ 10]

0.120

−0049 −0999 0.011 ¯ ∼(010) −0976 −0219 0.001 ¯ ∼(5¯ 10)

−0990 −0144 −0000 ¯ ∼[7¯ 10] 0.028 −1000 0.010 ¯ ∼[010]

0.159

0.018 −1000 −0002 ¯ ∼(010) 0.960 0.280 −0011 ∼(720) 0.056 0.996 −0075 ∼(010) −0978 −0206 −0013 ¯ 102) ¯ ∼(15

−0979 −0203 0.011 ¯ ∼[5¯ 10] −0097 0.995 0.003 ∼[010] 0.992 0.126 0.018 ∼ [810] 0.026 −0997 0.074 ¯ ∼[010]

0.164

−0383 −0810 0.455 ¯ ∼(1¯ 21) −0383 0.810 −0455 ¯ ∼(1¯ 2¯ 1)

−0379 −0801 −0452 ¯ ∼ [1¯ 2¯ 1] −0379 −0801 −0452 ¯ ∼[1¯ 2¯ 1]

0.052

0.120

0.159

0.164

0.170

0.170

0.52

(Continued)

368


Table 4.18. (Continued) Lattice invariant shear system (110) [001]

Habit plane indices (PI) −0178 −0047 1.006 ¯ ∼(106) −0855 0.483 0.195 ¯ ∼(952)

Direction of shape strain D1 −0859 0.494 0.134 ¯ ∼[741] −0132 −0076 0.966 ¯ ∼[107]

Magnitude of shape strain M1 0.115

0.115

Kelly and Ball (1986) and Kelly (1990) has pointed out that the grouping of martensitic variants which results in the formation of the important morphological features of zirconia martensites can be explained on the basis of habit plane predictions for different lattice correspondences. For both the correspondences B and C, a number of different LIS systems lead to two groups of habit plane. The first is in the region of (140)t for correspondence B (or (410)t for correspondence C). Depending on the correspondence adopted, the predicted orientation relationship is B-2 or C-2 within about 1 and in both the cases twin-related variants are predicted with a junction plane corresponding to (100)t . The twin relationship, though not precise, is normally less than 05 from an exact twin -relationship. The configuration of the twin-related variants illustrated in Figure 4.57 is produced for the B type lattice correspondence with the LIS on (011)t [211]t . It is to be noted that a similar configuration with two variants nearly twin related across the (100)m junction can also be generated when a different LIS system is chosen. The habit plane solutions for these cases are of the types {130}t and {140}t . The other group of solutions gives a habit plane which, for the correspondence B, is 2 or 3 from (100)t or (001)t and for the correspondence C is in the vicinity of (106)t (i.e. 3–12 from (001)t or (001)m ). The predicted orientation relationship is B-1 or C-1, respectively, and in both cases the theory predicts variants that are twin related about (001)m within 0.2–07 . An example of the twin-related configuration corresponding to the lattice correspondence B is schematically shown in Figure 4.57. A habit plane solution in the vicinity of (671)m or (761)m is predicted for a restricted choice of LIS systems, the magnitude of the shear being higher than those for the two sets of solutions discussed earlier. However, there is one significant exception, for correspondence C and for a LIS system of 110t ¯I12t a habit plane.

Martensitic Transformations (130)

T

369

or (14

0)T

16°

0) or (14 T (130) T

(a) Near (001)M i.e. (100)T (100)T//(001)M Near (001)M i.e. (100)T

(b)

Figure 4.57. The configuration of twin-related variants for B type lattice correspondence and ¯ t (after P.M. Kelly, 1990). LIS on (011)t [211]

4.7

TRANSFORMATION TOUGHENING OF PARTIALLY STABILIZED ZIRCONIA (PSZ)

Phase diagrams of binary oxide systems, discussed in Chapter 1, show that the two transition temperatures (corresponding to the cubic → tetragonal and tetragonal → monoclinic transformations) are lowered by alloying ZrO2 with CaO, MgO, Y2 O3 and rare earth oxides. Partial stabilization of ZrO2 , which results in a structure containing a mixture of the cubic and the monoclinic (or the tetragonal) phases, leads to an improvement in the thermal shock resistance and the toughness of zirconia ceramics. Reductions in the thermal expansion coefficient and in the volume change associated with the tetragonal → monoclinic phase transformation are factors which contribute towards the improved thermal shock resistance of PSZ (Subbarao 1981) (Table 4.19). The toughening, however, has been attributed Table 4.19. Lattice parameters (in nm) and thermal expansion coefficients of zirconia as a function of temperature. Temperature ( C)

am

bm

cm

956 (meas.) 0.51882 0.52142 0.53836 81 217 950 (calc.) 0.51881 0.52142 0.53835 81 22 Thermal 1.031 0.135 1.468 expansion coeff. (nm × 10− 6 C−1 )

Temperature ( C) 1152 (meas.) 950 (calc.)

at

ct

0.51518 0.52724 0.51485 0.52692 1.160 1.608

370


to a stress-induced martensitic transformation (Garvie et al. 1975, Porter and Heuer 1977, Evans and Cannon 1986, Kelly 1990). The martensitic transformation has two beneficial effects in this context. First, the transformation plasticity can partially relieve the applied stress; in other words, the energy of the transformation can directly contribute towards an increase in the fracture energy. Second, the strain associated with the transformed region may help counter the high tensile stresses at the tip of the advancing crack and resist its further propagation. Thermal processing of PSZ to achieve the maximum strength and toughness has been extensively studied and the results are summarized in several reviews (Pascoe et al. 1977, Nettleship and Stevens 1987, Evans and Heuer 1980, Subbarao 1981, Kelly and Rose 2002). The principle of microstructural engineering of PSZ for enhancing fracture toughness is based on distributing the tetragonal (and partly, the monoclinic) phase particles in the matrix of the cubic phase. The second phase may appear at grain boundaries during the sintering of zirconia, suitably alloyed for controlling the relative stabilities of the competing phases. Intragrain dispersion of the second phase occurs either during cooling from a temperature above the solvus line or during post-sintering heat treatments. Intragranular precipitates assume an ellipsoidal shape and are aligned along {100}c habits in order to minimize the strain energy of the precipitate-matrix assembly. The optimum size of the precipitate particles has been reported to be about 0.2 (m, as particles of still smaller sizes retain the tetragonal symmetry while larger particles transform into the monoclinic phase spontaneously. The role of metastable tetragonal precipitates in a cubic matrix in toughening PSZ has first been noted by Garvie et al. (1975) and later elucidated by Porter and Heuer (1979). It has been shown that all the precipitate particles within several micrometres of a crack possess a monoclinic symmetry, whereas all other particles are tetragonal. This suggests that the stress field near the crack tip causes the particles to transform into the monoclinic phase. By this mechanism, the elastic stress field at the crack tip can be dissipated through the formation of stress-induced martensite in the metastable tetragonal precipitates which also lose coherency in the process. Further extension of the crack, therefore, requires the application of an additional stress. Thus the stress-induced tetragonal to monoclinic transformation within the fine precipitates essentially operates as a crack blunting mechanism which is schematically illustrated in Figure 4.58. The retention in the fine (smaller than 0.2 (m) particles of the metastable tetragonal phase, when they are distributed in the cubic ZrO2 matrix, can be explained in terms of a much larger value of the strain energy associated with the cubic–monoclinic assembly in comparison with that associated with the cubic– tetragonal assembly (Porter and Heuer 1979). This strain energy difference more


371

Critical crack

Figure 4.58. Tetragonal to monoclinic transformation in zirconia with volume expansion resulting in microcracks around the particle; cracks propagating into the particle get deviated leading to increased fracture resistance. Table 4.20. Transverse rupture strength and fracture toughness of zirconia.

Tetragonal + cubic ZrO2 Monoclinic + cubic ZrO2 (overaged at 1400 C) Cubic ZrO2 (solution – annealed at 1850 C, 4 h)

Transverse rupture strength (MPa)

KI C (MN/m3/2

650 250

≈7.1 3.7

245

2.8

than compensates for the difference in the chemical free energy between the monoclinic and the tetragonal phases in the case of fine particles. A comparison of the strength and the fracture toughness of zirconia ceramics in three different microstructural states (Table 4.20) clearly indicates that the presence of the metastable tetragonal phase is directly responsible for enhancing the fracture toughness and transverse rupture strength quite substantially. Such a conclusion can also be arrived at from the observation that a nearly fully tetragonal material, prepared by fine particle technology and comprising a ZrO2 –Y2 O3 solid solution, shows a high strength at room temperature whereas a ZrO2 –CeO2 solid solution, in which the tetragonal phase is stable at room temperature, is relatively weak. The latter material exhibits enhanced strength and toughness at liquid nitrogen temperature, where the tetragonal phase is metastable. The phenomenon of toughening of PSZ by a stress-induced martensitic transformation can be compared with transformation-induced plasticity in TRIP steels, in spite of the fact that in the latter case the transformation occurs in the entire austenite matrix while in the former only the fine dispersed precipitates take part in the martensitic transformation (Figure 4.58).

372


4.7.1 Crystallography of tetragonal → monoclinic transformation in small particles The mechanism of transformation toughening described in the preceding section essentially requires the fulfilment of the following conditions: (1) The strain energy associated with the transformation should be of such a magnitude that particles below a certain critical size are retained in a metastable tetragonal state. (2) It should be possible to trigger a stress-induced martensitic transformation in these fine particles. (3) The volume change associated with the transformation should be positive so that it could counter the tensile stresses at the tips of advancing cracks. (4) The magnitude of the macroscopic shear associated with the transformation should be large for facilitating the occurrence of the stress-induced transformation. At the same time, ceramics being incapable of tolerating large shear strains, the overall shear strain associated with a single tetragonal particle should not be very high. These conflicting requirements can be met if a number of variants of the monoclinic martensite plates forming within a particle belong to a self-accommodating group. In that case, a major part of the shear strain associated with a single martensite variant can be neutralized by those associated with the other variants present in the group. The crystallography of the martensitic transformation in fine particles has been studied by Kelly (1990) mainly to identify the possible self-accommodating groups of martensite variants which can form within these fine particles (Figure 4.59). t mt m

[001]m [001]t

Strain = 4.12%

[100]m [100]t

Strain // [010]

Strain = 0.51%

= 0%

(a)

[001]m [001]t

Strain =0.97%

Strain // [010]

[100]m [100]t Strain = 2.65%

= 0%

(b)

Figure 4.59. The four possible arrangements of twin-related variants together with the range of strain values predicted for the distortion.


373

It is important to note that TEM studies on PSZ have established (Kelley 1990) that each of the fine tetragonal particles undergoes a stress-induced transformation, leading to the formation of an array of alternate bands, twin related on either (100)m or (001)m planes. These twins do not correspond to the LIS. Instead, these bands are twin-related variants which have shape strains that lead to the cancellation of the shear component when the two variants are of equal thickness. The large shear component associated with the formation of the first variant creates the “back stresses” which induce the formation of the twin-related variant with the opposite shear strain. The accommodation of shear strain by the combination of these twinrelated variants is responsible for the creation of the zig-zag arrangements of the variants.

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Chapter 5

Ordering in Intermetallics 5.1 Introduction 5.2 Theoretical Treatments 5.2.1 Alloy phase stability 5.2.2 Order–disorder transformations 5.2.3 The ground states of the Lenz and Ising model 5.2.4 Special point ordering 5.2.5 Concomitant clustering and ordering 5.2.6 A case study: Ti–Al system 5.3 Transformations in Ti3 Al-based alloys 5.3.1 → D019 ordering 5.3.2 Phase transformations in 2 -Ti3 Al-Based systems 5.3.3 Structural relationships 5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63 /mmc) and ordered orthorhombic (Cmcm) phases 5.3.5 Transformation sequences 5.3.6 Phase reactions in Ti–Al–Nb system 5.4 Formation of Zr3 Al 5.4.1 Metastable Zr3 Al (D019 ) phase 5.4.2 Formation of the equilibrium Zr3 Al (L12 ) phase 5.4.3 +Zr2 Al → Zr3 Al peritectoid reaction 5.5 Phase Transformation in -TiAl-Based Systems 5.5.1 Structural relationship between 2 - and -phases 5.5.2 Phase reactions 5.5.3 Transformation mechanisms 5.6 Site Occupancies in Ordered Ternary Alloys 5.6.1 Ordering tie lines 5.6.2 Kinetic modelling of B2 ordering in a ternary system 5.6.3 Influence of binary interaction parameters 5.6.4 B2 ordering in the Nb–Ti–Al system References

380 383 384 386 397 401 407 412 416 416 417 421 424 428 432 436 437 439 441 443 443 446 451 458 458 460 462 464 465


Chapter 5

Ordering in Intermetallics

List of Symbols k ij : Short-range order parameter for the ij-pair in the kth coordination shell B : Bulk modulus of a superstructure EF : Fermi energy k : The kth volume expansion coefficient for Etot Etot : Total electronic energy of a superstructure Ecoh : Cohesive energy of a superstructure : Formation energy of a superstructure Eform : A superstructure : A cluster : Long-range order parameter Jt : Effective cluster interaction for the cluster of type t jk : The kth volume expansion coefficient for J

k: A wave vector in the reciprocal space kB : Boltzmann constant NEF : Density of states at the fermi level r: Atomic occupation probability at position r pni : Site occupation operator for a species i at a site n k: Amplitude of the concentration wave with wave vector k Probability of a -point cluster to occur in configuration

: i : A site operator which can take value +1−1, for a component A (B) in a binary alloy Ti− : Instability temperature below which the disordered phase is unstable w.r.t. spontaneous ordering Ti+ : Instability temperature above which the ordered phase is unstable w.r.t. spontaneous disordering Teq : Equilibrium temperature T0 : Temperature corresponding to equal free energies of two phases Tcs : Conditional spinodal temperature for the ordered phase V0 : Equilibrium volume of a superstructure 379

380


t : wij : G: F: H: G: H: kF kR :

F R :

5.1

Correlation function for a -point cluster of type t Interaction potential between ‘ij pair of atoms Gibbs free energy Helmholtz free energy Enthalpy change for a given process Space group of a given crystalline phase Subgroup of a given space group G Rate constants for the forward and backward reactions, respectively Pre-exponential factors for the forward and backward reactions, respectively

INTRODUCTION

In an alloy which can be approximated as an ideal solid solution, the distribution of different atomic species in the lattice is nearly statistical. The interaction energies, Vij , between atomic pairs, ij, are such that there is no preference for the formation of bonds between either the like or the unlike atoms. When a solid solution exhibits a tendency to deviate from the ideal behaviour, one can visualize two alternatives. The like atoms tend to cluster together when there is a preference for the formation of bonds between like atoms. The second alternative is that the atoms arrange themselves in an ordered array where specific atomic species occupy specific lattice sites. The clustering and ordering tendencies are, respectively, associated with a positive and a negative deviation from ideality, as the free energy of the respective systems deviates in the positive or in the negative direction from the free energy of the ideal solid solution. The phenomenon of ordering in an alloy can be described in terms of the simple example of the ordering of a bcc (A2) structure to a CsCl type (B2) superlattice (Figure 5.1(a)). Let us consider an equiatomic alloy of the elements A and B in which the atoms are distributed in the bcc lattice. Considering only the nearest neighbour interactions, one can intuitively see that if VAB , a measure of the attractive interaction between the unlike atoms, is much stronger than the interaction between like atoms, VAA and VBB , there will be a tendency for A and B atoms to, respectively, select the body centre and the corner positions of the unit cell of the bcc lattice or the other way round. In case a perfect order is established, all A atoms will have only B atoms as their first near neighbours and vice versa. When the temperature of such an alloy is raised, the entropy factor plays an increasingly important role and above the order–disorder transition temperature, Tc , the distribution of atoms in the lattice sites becomes random. The competition between


B32

B2 (a)

381

D03 (b)

(c)

Figure 5.1. bcc-based ordered intermetallic phases which are the ground state superstructures under the first and the second nearest neighbour (NN) pair approximation.

the entropy factor, which tries to randomize the system at elevated temperatures, and the enthalpy factor (resulting from interatomic interactions), which tends to organize the atoms in an ordered array, dictates the state of the order. When the formation enthalpy of an ordered structure is very large, the state of order may persist even up to the melting temperature, and in such cases an order to disorder transition is not encountered in the solid state. Ti- and Zr-based alloys are essentially based on the hcp and bcc lattices, corresponding to the two allotropic modifications of these elements. Studies on the order–disorder transition in these alloys, therefore, include an examination of the possible superlattice structures based on the bcc and hcp lattices, the determination of ground states (energetically favourable superlattice structures at 0 K) and investigations regarding the mechanism of order evolution in these systems. The interest in order evolution is centred around the question as to whether the process occurs in a homogeneous manner by the development of a concentration modulation with appropriate wave vectors, followed by a gradual amplification of such modulations or by a heterogeneous process involving the nucleation of nearly perfect ordered particles in the disordered matrix, followed by the growth of these particles. In the context of Ti and Zr alloys, the disorder–order transition has been studied in detail in only a few systems. The most important among the structural transitions involved in these are: (a) (hcp)→ D019 in the Ti–Al, Ti–Sn and Ti–Ga systems in the vicinity of the Ti3 X (X = Al, Sn, Ga), and (b) (bcc) → B2 in the Ti–Al and Zr–Al systems. Theoretical treatments for the determination of thermodynamic parameters of ordering processes and experimental results which illustrate the essential features of these processes have been reviewed in this chapter.

382


Phase transformations (including phase reactions) involving ordered phases in the aluminides of Ti and Zr have attracted attention ever since the emergence of these intermetallics as possible structural materials. Chronologically Zr3 Al was one of the first intermetallics to be considered as a structural material. The prospect of using stoichiometric Zr3 Al as a pressure tube material in pressurized heavy water nuclear reactors (PHWRs), originally conceived by Schulson (1977), was quite bright because of the exceptionally good resistance of this material against irradiation creep, which happens to be one of the life-limiting factors under the prevailing service conditions. The presence of atomic order could be regarded as being essentially responsible for the superior irradiation creep resistance of this material. The intermetallic was designed on the basis of the maxim that ordering generally reduces the rates of migration of vacancies and interstitials, thereby enhancing the probability of annihilation of these defects. This, in turn, reduces the irradiation creep rate under a given stress. Some other virtues of this material, namely, a low thermal neutron absorption cross-section, promising mechanical properties and good corrosion resistance in water in the temperature, pressure and radiation environment present in the PHWRs make it eminently suitable for the proposed application. The reason for which Zr3 Al was not finally accepted for pressure tube application was the tendency of disordering, leading even to amorphization, of this alloy under irradiation and its poor room temperature ductility. The equilibrium structure of Zr3 Al is L12 (Cu3 Au type) and this phase evolves through a peritectoid reaction. A classical order–disorder transition cannot be studied in this system as Zr3 Al cannot be congruently disordered, unless radiation disordering is resorted to. Several interesting aspects of phase transformations, such as the peritectoid reaction between the and Zr2 Al phases to form Zr3 Al, the formation of a metastable Zr3 Al phase (D019 structure) during precipitation from the supersaturated (hcp) matrix of non-stoichiometric Zr–Al alloys and the cellular reaction leading to the formation of an + Zr 3 AlL12 lamellar aggregate, have been discussed in this chapter. The story of the development of Ti aluminides is even more exciting. The need for the development of suitable aerospace materials for use in the temperature range of 770–920 K has been felt for bridging the gap between the temperature ranges pertinent to conventional high -Ti alloys on one hand and Ni-based superalloys on the other. The binary intermetallic Ti3 Al (2 -phase) has a specific modulus and a stress rupture resistance comparable to those of superalloys but the complete absence of ductility at room temperature comes in the way of its being accepted as a structural material. Blackburn and Smith (1978) were the first to demonstrate room temperature ductility in a Ti3 Al-based composition (Ti–24% Al–11% Nb) and this was followed by wide-ranging investigations on phase transformations


383

in and deformation behaviour of a large number of Ti aluminides. These studies have revealed a variety of phase transformations in multicomponent Ti aluminides, one of the most exhaustively studied systems being the ternary Ti–Al–Nb. The constituent phases, the sequences and mechanisms of transformations and the mechanical properties are quite distinct in different composition regimes for the Ti–Al–Nb system. Based on these considerations, the transformation processes can be discussed in two distinct groups of alloys, one based on Ti3 Al and the other based on TiAl ( phase, L10 structure). The Ti3 Al-based alloys can again be categorized on the basis of their -stabilizer content. Important alloys belonging to the first category are Ti–24Al–11Nb (Blackburn and Smith 1978) and Ti-25Al-8Nb-2Mo-2Ta (Marquardt et al. 1989). They contain -stabilizers in the range of 10–12 at.%. Alloys with about 14–17 at.% -stabilizers, including Ti-24Al-(14-15)Nb (Blackburn and Smith 1978) and Ti24Al-10Nb-3V-1Mo (Blackburn and Smith 1989), can be grouped into the second category. The third category of alloys, containing 25–30 at.% -stabilizers, has been developed more recently (Rowe 1991, Gogia et al. 1993). Phase reactions and mechanical properties associated with these three categories of Ti3 Al-based alloys are quite different and, therefore, the development of microstructure in these groups of alloys has been discussed separately. The phase equilibria and solid-state phase transformations in -TiAl alloys, with the Al content varying between 40 and 50 at.%, have also attracted considerable attention in the recent years. This is primarily because of the fact that -TiAl-based alloys offer an attractive combination of properties, namely, low density, substantial high temperature strength and good creep as well as oxidation resistance. These alloys, at room temperature, usually consist of the 2 (D019 ) and (L10 ) phases. The 2 -phase disorders into the hcp -phase above about 1420 K. The occurrence of the eutectoid reaction → 2 + , the existence of unique orientation relationships connecting the three phases and the sensitivity of phase transformation modes to the cooling rate are some of the factors responsible for the development of many interesting microstructures in -TiAl alloys. An attempt has been made in this chapter to rationalize the variety of microstructures reported in -TiAl alloys in terms of some plausible phase transformation schemes.

5.2

THEORETICAL TREATMENTS

In this section, we will present some state-of-the-art theoretical tools to study phase stability and ordering in alloys. Since most commercial alloys are complex multiphase mixtures, it is important to have knowledge of all the possible alloy phases that can occur in a given system: terminal solid solutions, possible ordered

384


intermetallic phases and other disordered solid solutions. Hence, the study of alloy phase stability and order–disorder transformations is a major area of research today. 5.2.1 Alloy phase stability More than 70 years ago, Hume-Rothery (1967) laid the foundations for the study of alloy phase stability in relation to the electronic structure. He pointed out the connection between the observed crystal structures and the electron concentration, which is related to the valence electron difference between the constituent elements and to the composition. Earlier theories of stability of alloy structures have identified the important role of the density of states (DOS) curve for correlating the extent of phase stability with the electron concentration (e/a). The DOS is defined as the number of electronic states per atom (or volume) with energies between certain values: NEo =

dS 3 4 E=Eo Ek

(5.1)

where the integration is carried out over a constant energy surface in k-space between Eo and Eo + E and is the cell volume. The first attempt in this direction was made by Jones (1962) in terms of relative electronic DOS curves between any two simple alloy phases (1 and 2 ). He tried to explain the phase competition between the - and the -phases in the Cu–Zn system in terms of a rigid band model for the conduction electrons. The gist of the argument was that, within the low e/a range corresponding to the -phase, the total DOS for the -structure is higher than that for the -structure which follows it, because of the prominent peak in the DOS curve associated with the 111 zone contact with the Fermi surface. Thus the conduction electrons can be accommodated within a lower total energy than that for the -phase. As e/a increases, the 110 peak in the DOS curve of the -phase structure becomes prominent, making it a more favourable structure. Thus, at alloy compositions at which the Brillouin zone boundary is just touched, a given crystal structure should be particularly favoured. These compositions correspond to those of Hume-Rothery phases. Experiments have proved that in many metals, the Fermi surface is very different from that of a free electron sphere, giving birth to the nearly free-electron theory. The deep potential energy wells at the centres of atoms play a very small role. In 1959, the pseudopotential theory (Harrison 1966, 1973) took roots; in this, it was considered that as a conduction electron passes through the central region of an ion in a metal, it not only experiences a strong electrostatic attraction to the nucleus but also a strong repulsion from the core electrons. In some metals


385

like Na, Mg and Al, these two opposite contributions almost cancel, so that the ions in these metals appear to be nearly transparent to the conduction electrons. It thus becomes possible to replace the true potential in the Schrödinger equation by a pseudopotential in which the Pauli repulsion is introduced to cancel out most of the coulomb potential of the nucleus (which can be treated using perturbation methods on the free electron theory). The present status of the electronic structure methodologies is attributed to three important breakthroughs during the last 30 years or so. The first was the introduction of Hohenberg–Kohn–Sham density functional theory (DFT) (Hohenberg and Kohn 1964, Kohn and Sham 1965) in the mid-1960s, when it was rigorously proved that the complicated and unmanageable many-electron problem can be transformed into an effective one-electron problem which, in turn, can be solved within the so-called local density approximation (LDA) (Jones and Gumarsson 1989). The DFT is based on the quantum statistical approach. It does not attack the many-body problems frontally, but it possesses a certain conceptual simplicity for which it has emerged as the most successful tool in describing the ground state properties of inhomogeneous electronic systems. The second breakthrough came in 1975, when linear methods for solving the one-electron band structure of solids was introduced by Andersen (1975). The LAPW (linear augmented plane wave) (Andersen 1975, Jansen and Freeman 1984) and the LMTO (linear muffin-tin orbital) (Andersen et al. 1993) methods and associated methods like the ASW (augmented spherical wave) method are the most widely used today for ab initio (i.e. without any a priori assumption regarding interatomic interactions) investigation of materials. The third breakthrough came in 1985, when Car and Parrinello (1985) proposed a recipe for ab initio molecular dynamics (MD) simulation of atomic aggregates, where “forces” acting on atoms are evaluated within the framework of DFT. This DF-MD method has since been extensively used for studying various kinds of clusters, nanoparticles, liquids and amorphous solids, although the underlying plane wave pseudopotential makes its applications restricted mainly to s–p bonded systems (Payne et al. 1992). In contrast to ordered solids, the calculation of any physical property of a disordered alloys requires configuration averaging over all possible configurations. Most experiments measure configurationally averaged properties. The configuration averaging is usually achieved in the framework of the coherent potential approximation (CPA) (Soven 1967, Taylor 1967). The essence of the CPA is to replace the array of random potentials by an effective energy-dependent coherent potential (z). The scattering properties of z are then determined self-consistently (in a single site mean-field sense), from the requirement that an electron travelling in an infinite array of z undergoes, on average, no further scattering upon replacement of any z with the actual potential. So CPA is a

386


self-consistent prescription which allows one to obtain an effective non-random Hamiltonian. CPA, coupled with first principles band structure methods in the framework of LDA, provides a starting point for the ab initio calculation of the electronic structure of disordered alloys. Though CPA provides reliable results, there are many situations where the single site approximation inherent in CPA begins to fail: as in cases where clustering effects become important (e.g. in the impurity bands of split band alloys, like the Zn band in a Cu-rich CuZn alloy), where short-range order dominates leading to ordering or segregation and where local lattice distortion because of size mismatch of the constituents leads to essential off-diagonal disorder (as in CuPd and CuBe alloys). Recently the augmented space recursion (ASR) method which involves the recursion (Haydock 1980) method in the augmented space (Hilbert space ( ) + Configurational space ()) (Mookerjee 1973a,b, 1987) is gaining importance to handle disordered alloys. The recursion method (Haydock 1980) offers an alternative to the band structure method. This method is suitable for computing the local properties which are related to the diagonal elements of the resolvent, where it offers a large computational advantage over the band structure methods. Moreover, the recursion method is based on real space and so does not require lattice periodicity for its operation, in contrast to the band structure methods. As a result, both ordered and disordered systems as well as systems with broken translational symmetry (such as surfaces) can be treated within the recursion method. The augmented space formalism, introduced by Mookerjee (Andersen et al. 1993), is a novel and conceptually attractive method for the calculation of the configurationally averaged Green function of a disordered material. In this method, one transforms the Hamiltonian describing a given disordered system to an ordered Hamiltonian whose Green function matrix elements correspond to appropriate configurational averages of the Green function of the original disordered system! The new ordered Hamiltonian is said to be in an augmented space which can be described as the direct product of the Hilbert space, , spanned by the original Hamiltonian with a configuration or disorder space, , which spans all possible configurations of the system. 5.2.2 Order–disorder transformations The thermodynamics of solid solutions consists of contributions from electronic, displacive, vibrational, magnetic and configurational degrees of freedom. The theoretical model presented here assumes that the configurational degree of freedom can be separated from the other degrees of freedom and that change in the latter effects can be neglected in certain types of phase transformations or phase changes, called coherent phase transformations. These coherent phase transformations are


387

defined as those where the basic lattice framework of the crystalline solid solution is conserved, i.e. the two phases differ from each other only in the chemical distribution of the constituent elements. Hence chemical rearrangement of atomic species is allowed along with displacements from the average lattice positions, but any discontinuous change of the lattice framework, leading to dislocations, incoherent interfaces, free surfaces and grain boundaries, is not allowed. 5.2.2.1 Historical developments As far back as 1925, Lenz and Ising (Temperley 1972) introduced a model to study the thermodynamics of magnetic systems. They proposed to associate a magnetic moment at each lattice site of a given lattice. These magnetic moments can have only two components, up or down, along a given direction, and a moment interacts with only its nearest neighbours. The pairs of parallel moments have an energy −J , whereas pairs of antiparallel moments have an energy +J . It is then quite easy to calculate the energy of a given configuration of a finite system. It is not always possible to obtain explicit expressions for the thermodynamic quantities of interest when the size of the system increases to infinity. Various approximations have been proposed for the solution of the Lenz and Ising model in three dimensions (3D). The resemblance of this model to a binary substitutional alloy in which each site is occupied by either the A or the B component was immediately evident, with a site-occupation operator replacing the magnetic moment. The same analogy can be extended to interstitial alloys where each interstitial site is either occupied or vacant, disregarding the host matrix. This model, within certain approximations, has been shown to contain the essential ingredients to account for various types of phase transitions and various topologies of phase diagrams. The thermodynamic state of a system can be easily obtained by finding the proper expression for the partition function = exp−E/kB T, where signifies a configuration and the summation is over all possible configurations. The basic problem is then to calculate the partition function and its derivatives with respect to external fields, from which we can obtain the free energy and all averages of interest. Except in a few cases, it is not possible to perform these calculations. One has, therefore, to make some plausible approximations. The most useful ones are the so-called mean field (or molecular field) approximation (MFA) (de Fontaine 1979, Ducastelle 1991). Mean field theories (MFTs) are, in general, derived from variational principles and have been shown to suffer from serious drawbacks, particularly close to second order phase transitions, which are characterized by the fact that long-range order (LRO) parameter is a continuous function of temperature, vanishing at the critical temperature, Tc . At temperatures near Tc , the singularities of the thermodynamic functions are not correctly reproduced. In such cases, it is necessary

388


to treat the fluctuations in the medium. The celebrated renormalization group theory (Ma 1976, Toulouse and Pfeuty 1977) has been developed in the 1970s to account for this critical regime. This theory is specially designed to identify the universal characteristics of phase transitions. However, at temperatures away from Tc , its practical implementation is not very easy and also not very reliable. Since most of the order–disorder transitions concerning the 3D Lenz and Ising model seem to be of the first order, i.e. the LRO parameter is discontinuous at Tc , transition occurs before the fluctuations have increased too much in the disordered phase. In such a situation, MFTs are expected to give reasonably accurate results. The simplest MFT (called the Bragg–Williams approximation) (Bragg and Williams 1934, 1935, Wilson and Kogut 1974), applied to the Lenz and Ising model, easily predicts the occurrence of order–disorder transitions and provides a description of the disordered and ordered states. This is a single-site approximation in which each atom is assumed to be embedded in a mean field. Due to its simplicity, this theory is successful in many respects, but quantitatively it is rather inaccurate. The critical temperature are found to be much too high, the short-range order is rather badly treated, and the phase diagrams can even be qualitatively wrong in the case of frustrated lattices. In this theory, correlation between sites or the local statistical fluctuation was totally neglected. In view of this defect, Bethe (1935) introduced the concept of short-range order. This is to improve the approximation from a single site in the mean field to a pair of sites in a molecular field. As expected, a great difference from the result of the B–W approximation appears near the critical point. In particular, the results showed the existence of short-range order above Tc , which decays as the temperature is raised. In addition, the critical point was lowered in comparison to the B–W approximation for the same value of pair interaction, J2 . Generally the better is the degree of approximation, the lower is the Tc , and thus, the Tc is taken as a measure of the degree of approximation. Since the exact solution of the 3D Lenz and Ising problem cannot be obtained, the most important source of exact information concerning this problem lies in a series expansion of the partition function and other derived properties at low and high temperatures. An important example of a high-temperature series expansion is due to Kirkwood (1938), who devised an ingenious method to evaluate the partition function in a systematic way by a series expansion in terms of the moments of probability distributions. Kirkwood evaluated these moments down to the third order. The series expansion approaches can be classified into two types. In the first, a particular selection of terms which can be readily evaluated is summed to provide closed form approximations similar to other approximate methods. In this


389

connection, the method of Rushbrooke and Scions (1955) and that of Yvon (1948) and Fournet (1967) may be cited. In the second kind of approach, the known terms of the expansion are used to assess its asymptotic behaviour. This approach enables one not only to estimate the true critical point but also to closely investigate the critical behaviour. These methods involve cumbersome mathematical calculations, without adding much physical insight. 5.2.2.2 Static concentration wave model Landau and Lifshitz (1969, 1980) developed some very useful concepts related to order–disorder transitions based on the MFT. Since their theory was based on abstract group theoretical arguments, it could not be exploited by the metallurgists initially. However, these concepts have been widely used by some Russian scientists like Krivoglaz (Krivoglaz and Smirnov 1964) and Khachaturyan (1978, 1983; static concentration wave (SCW) model). Elsewhere the theory has been applied mostly to displacive soft mode transitions. The SCW model is in fact a B–W model in an arbitrary number of LRO parameters and has been derived from Landau’s general analysis of order–disorder reactions. It is useful for analysing order–disorder systems in which the ordered phases are crystallographic derivatives of their respective parent phases (coherent phases). In the SCW model, the crystal is modelled by a rigid lattice onto which “average atoms”, or atoms whose properties are assumed to be averaged according to the occupancy of particular sites, are superposed. The internal energy is assumed to be derived only from pairwise interactions between atoms, and the entropy is taken to be equal to the sum of the entropies of the average atoms. In this model, the occupation probability, (r), at a given site, r(p), is expanded in a Fourier series, i.e. it can be represented as the sum of SCWs whose amplitudes are the Fourier coefficients and whose wave vectors determine the superstructure period: r =

N

ke−ikhrp

(5.2)

h=1

where (k)s are the Fourier coefficients given by k =

N 1 reikhrp N p=1

(5.3)

The first summation is over the N lattice points of the periodic crystal and the second is over the N points of the first Brillouin zone. The wave vector k(h) and the lattice vector r(p) are defined by

390


rp = p a ( = 1 2 3; p are integers, summation implied) kh = 2h b (h = m /N , m = 0, ±1 ±2 ) where a and b are lattice translation vectors and primitive translation vectors of the reciprocal lattice such that a · b = . The concentration wave amplitudes k corresponding to the wave vector k that generates the ordering instability are expressed in terms of normalized (with respect to c) LRO parameter, n , via the following relation, k = n c

(5.4)

The normalized order parameter ( n ) is related to the standard order parameter ( ), as n = / max , where max is the maximum order parameter attainable at a given composition. For a disordered phase, (r) is independent of r and is equal to the atomic fraction, c, of the solute. As an example, we will briefly illustrate the binary L12 ordering in an fcc solid solution using the SCW model. The L12 structure is one in which all the facecentred positions are occupied by one type of atoms and all the corner positions are occupied by the other type of atoms. For the L12 phase, all the (k) values vanish, except (000), (100), (010) and (001) (Figure 5.2). The value of (000) is just the bulk (or solute) concentration, c, and the other three amplitudes are equal. The occupation probability for each sublattice can then be calculated to give ∗ ∗ ∗ r = c + e−i100 r + e−i010 r + e−i001 r

(5.5)

where = 100 = 010 = 001 and 000 = c. The exponents in the above equation have only two values: −i and −i2 depending on whether the position vector r lies in a 200 or a 100 plane with respect to the origin, respectively. The atoms are situated on the fcc lattice such that they lie either (a) in all three 100 planes or (b) in one 100 and two 200 planes, so that (r) assumes only two values: (1) c + 3, if the atom is in a corner position and (2) c − , if it is in a face-centred position. There are 41 N corner atoms and 43 N face-centred atoms. The internal energy, in the pair approximation up to an arbitrary coordination shell, is given by E=

N N Jkhkh∗ kh 2 h=1

(5.6)


391

(001)

(100)

(010)

Figure 5.2. The L12 structure (AB3 stoichiometry) viewed as superposition of three {100} concentration waves. Filled (open) circles represent A (B) atoms, respectively.

where the star (∗) indicates the complex conjugate of the amplitude of the corresponding concentration wave, and Jk, the Fourier transforms of the pair interactions, are given by1 Jk = 1/N

N

Jreikhrp

(5.7)

p=1

Hence, for the L12 structure, the internal energy is E=

N J000c2 + 3J1001002 2

(5.8)

The expression for the configurational entropy is given as S = kB

N

rp ln rp + 1 − rp ln1 − rp

p=1

1

The dependence of r and k on p and h are assumed if not shown explicitly.

(5.9)

392


which, in terms of sublattice probabilities, can be expressed as S = kB

=1

NS ln + 1 − ln1 −

(5.10)

where is the total number of sublattices, while and NS are the occupation probability and the number of atoms on the th sublattice, respectively. In the SCW model, the internal energy is already given in terms of the order parameter (Eqs. (5.6) and (5.8)) or the amplitude of the concentration wave. The LRO parameter can, most comfortably, be substituted in the expression for the entropy, as will be seen now. The expression for the configurational entropy, after substitution of Eq. (5.5), is 1 S = −NkB c ln c + 1 − c ln1 − c − c − k lnc − k 2 1 (5.11) + c + k lnc + k 2 where kB is the Boltzmann constant. Hence the Helmholtz free energy, F , given by F = E − TS

(5.12)

is obtained as a function of T c and on substituting the normalizing relation (5.4). The free energy of the L12 ordered phases is given as F L12 T c n =

1 k T J000 + 3Jk 2n c2 + B 3 c1 − n ln c1 − n 2 4 + 3 1 − c1 − n ln1 − c1 − n + c1 + 3 n lnc1 + 3 n + 1 − c1 + 3 n ln1 − c1 + 3 n

(5.13)

5.2.2.3 Cluster variation method The most powerful approximation, now, is the cluster variation method (CVM) (Kikuchi 1951, Sanchez and de Fontaine 1978, Morita 1984, Fontaine 1994) which includes the correlation between a few sites in some clusters. The CVM of statistical mechanics was originally proposed as an improved approximation for the solution of the Lenz and Ising model. It is, in fact, a hierarchy of approximations


393

ranging from the simplest Bragg–Williams–Gorsky single-site (one-point) approximation to a very elaborate one considering clusters of lattice points of varying size, shape and complexity. Therefore the CVM is regarded as a natural way of generalizing the MFT for the solution of the 3D Lenz and Ising model used for the study of configurational thermodynamics of alloys. In the CVM, the statistical thermodynamics of alloys is described in terms of atomic configurations of clusters of lattice sites. The state of partial order in alloys is described in terms of multisite correlation functions, as will be described below. Detailed accounts of the subject are provided in many books and review articles such as those due to Ducastelle (1991), de Fontaine (1979, 1994), Morita et al. (1994), Inden and Pitsch (1991), Mohri et al. (1985). The linkage between quantum mechanics and statistical mechanics leading to the first principles configurational thermodynamics of alloys comes via the configurational energy expression which is expanded in terms of effective pair/multisite (cluster) interactions (ECI). These configurationally averaged effective interactions are calculated using ab initio methods. Traditionally there have been two approaches for obtaining these ECIs. The first approach is to start with the electronic structure calculations and the total energy determination of some judiciously selected ordered superstructures (configurations) of a given alloy system and expand the energy in terms of ECIs. The ECIs can then be calculated by the inversion of the expansion – the Connolly–Williams inversion method (IM) (Connolly and Williams 1983, 1984). The other approach is to start with the disordered phase, set up a perturbation in the form of concentration fluctuation and study whether the alloy can sustain such a perturbation. This approach includes the generalized perturbation method (Ducastelle and Gautier 1976), the embedded cluster method (Gonis et al. 1984) and the concentration wave approach (Gyroffy and Stocks 1983), all of which are based on the CPA. The direct configuration averaging (DCA) method (Dreyssé et al. 1989) consists of obtaining the effective pair interactions (EPI) for each random configuration of the disordered alloy using the recursion technique and then performing the configurational averaging over a (selected) number of configurations. In the ASR (Mookerjee 1973a,b) along with the orbital peeling (OP) (Mookerjee 1987) technique, the recursion is applied in the augmented space and the averaging is performed over all the possible configurations of the disordered alloy to obtain the EPIs. The present treatment is based on the article by Inden and Pitsch (1991). Consider a crystalline system with N lattice sites and having M constituents. These constituents can be atomic species like A, B, C (or can be “vacancies” also) such that i Ni = N , where Ni is the number of atoms of type i. A particular distribution of these constituents on the lattice sites defines a configuration. A configuration is specified by site operators, n (n = 1 2 N ), which may take

394


values (numerical labels) i (i = 1 2 M), corresponding to the occupation of the site n by the constituent i. For a M-component system, the site operator can take the values M − 1 0 −M − 1, e.g., for a binary system, n corresponding to the lattice site n can take the two values ±1, depending on whether the site n is occupied by component A (+1) or by component B (−1). Any configuration is then specified by a N -dimensional vector = 1 2 N . In total, there are M N different configurations, as each of the N sites can be occupied in M different ways. Any function of n (including n itself) is called a configurational variable. A second operator pni , called the site occupation operator, which allows to count the number of sites n, occupied by the same type of atom i, for taking averages, is defined as follows: pni

=

1 0

if an atom of type i occupies site n otherwise

The cluster approximation: It is much more convenient to consider the configurations of much smaller units called clusters in place of the N -point lattice which is quite large. A cluster is defined by a set of lattice points 1, 2, …, and a configuration on this cluster is given by = 1 2 (Figure 5.3). On a -site cluster there are, in total, M M M N ) configurations. The configurations of the N -point system can then be classified into groups with the same number of clusters N with the configuration . The probability, , of a -point cluster to occur in a configuration is given by = N /N

(5.14)

b a

c

d

Figure 5.3. The irregular tetrahedron (IT) cluster (abcd) approximation for the bcc lattice.


395

where N is the number of equivalent -site clusters contained in the system. These fractions are called cluster probabilities or reduced density matrices. They specify the configuration in the -point cluster approximation. The thermodynamic functions derived by the CVM depend on the probabilities of this cluster and also on the probabilities of all the subclusters which can be derived from that of the largest cluster by partial summations. The correlation functions: We characterize the atomic configuration in terms of correlation functions (s) by considering averages over equivalent clusters. An -point ( ≤ ) correlation function, i , for the cluster type i, can be defined as i =

1 i

N

1 2 · · ·

(5.15)

where the summation is taken over all the possible configurations of the cluster of type i. These correlation functions, for a perfectly ordered state, can be determined by inspection. However, these correlations serve as variables with respect to which the free energy of the system is minimized to arrive at the equilibrium state of order (or partial order). In order to determine the stability of a configuration at T = 0 K and in the equilibrium with respect to exchange of atoms, one has to minimize the internal energy, E G T V !A !B , in the grand canonical scheme, given as a function of temperature (T ), volume (V ) and chemical potential (!i ). The grand canonical internal energy (E G ) is obtained from the canonical energy, E, by its Legendre transformation: E G T V !A !B = ET V NA NB +

M

!i Ni

(5.16)

i=1

Generally, one considers a constant volume system and the configurational part of the internal energy is separated. The configurational energy, EcG , in the simplest approach, is expressed in terms of pairwise interactions (Vij ) up to an arbitrary coordination shell as 1 k k Z Vij EcG = N 2 k ij

ij 0k + N

!i

i 0

(5.17)

i

where Zk is the coordination number for the kth coordination shell and position “0” stands for any position in the crystal taken as the origin. The CVM entropy (S) is evaluated according to S = kB ln , where is the number of arrangements which can be formed for given values of the correlation

396


functions. In the CVM, we take correlations only up to the basic cluster size () and therefore, the thermodynamic probability, , is the number of possible arrangements of this basic cluster, corrected to take account of the overlapping clusters. We will not derive the expression for the CVM entropy here. Interested reader is referred to some of the excellent articles mentioned above. In brief, the CVM entropy takes the form S = −kB N

=1

m a

ln

(5.18)

where the sum runs over all the subclusters, , of the basic cluster, . The number of -clusters per lattice point, m , and the Kikuchi–Barker coefficients, a , are to be derived by geometrical considerations of the lattice. The cluster probabilities,

, can be expressed as

1 (5.19) v

= 1+ 2

≤ where the summation is over all the subclusters, , of the cluster, ; v are a sum of -order products of spin variables , the structure of which depends upon the symmetry of the cluster in question. As an example of what has been discussed above, we briefly present here the CVM treatment for the equiatomic B2 (CsCl type) structure which is a bcc-based superstructure (see Figure 5.1). The B2 structure has two sublattices corresponding to the corner and the body-centred positions, respectively. The cluster approximation selected is irregular tetrahedron cluster as shown in Figure 5.3. Let us represent the basic cluster tetrahedron by (abcd), denoting the four vertices. In a perfectly ordered B2 structure, the cluster sites (a) and (b) will be occupied by type “A” atoms ( 1 = 1) and sites (c) and (d) will be occupied by type “B” atoms ( 2 = −1). Now the enumeration of different subclusters of this basic cluster gives: (1) (2) (3) (4) (5) (6) (7) (8)

point cluster (type 1): a/b point cluster (type 2): c/d NN pair cluster (type 1): (ac), (bc), (ad) and (bd) NNN pair cluster (type 2): (ab) NNN pair cluster (type 3): (cd) isosceles triangular clusters (type 1): (abc), (abd) isosceles triangular clusters (type 2): (acd), (bcd) irregular tetrahedron cluster: (abcd).


397

The subcluster probabilities corresponding to the above clusters are then given as (here we have used short notations: i ≡ a , j ≡ b , k ≡ c and l ≡ d ), 1 1

= 21 1 + i1

2 1

= 21 1 + k1

1 2

= 41 1 + i1 + k1 + ik2

2 2

= 41 1 + i + j1 + ij2

3 2

= 41 1 + k + l1 + kl2

1 3

= 18 1 + i + j1 + k1 2 + ik + jk2 + ij2 + ijk3

2 3

= 18 1 + j1 1 + k + l1 + jk + jl2 + kl2 + jkl3

1 4

=

1

2

1

2

1

2

1

2

3

1

1

2

2

1

3

1

2

1 2 1 2 3 1 1 + i + j1 + k + l1 + ik + il + jk + jl2 + ij2 + kl2 16

+ ijk + ijl3 + ikl + jkl3 + ijkl4 1

2

1

On substituting the above expressions for cluster probabilities in the CVM free energy expression (Eqs. 5.17 and 5.18), one obtains free energy (F = E − TS) as a function of T , !i s and set of eight independent correlation functions with respect to which the F is to be minimized. 5.2.3 The ground states of the Lenz and Ising model The determination of ground states of the Lenz and Ising model is much easier than evaluating the free energy of a system at finite temperatures. Many exact results have been obtained (Allen and Cahn 1972, de Fontaine 1979, Sanchez et al. 1982, Kanamori and Kaburgi 1983, Ducastelle 1991) when the interactions are short range. This is based on a method of linear inequalities that allows the determination of the most stable ordered structures as functions of the concentration and the strength of the pair interactions. In practice, it is very difficult to include more than fourth or fifth nearest neighbour interactions. If the range of interactions is finite, only a finite number of ordered structures can be found. The correlation functions characterizing the atomic configurations can only vary within a restricted range of numerical values due to certain consistency conditions. This restricted range of values is called the existence domain or the

398


configuration polyhedron outside which the values do not define actually existing atomic configurations. The dimensions of the configurational space depend either on the number of correlations or equivalently on the size of the largest cluster. These existence domains are most useful for an analysis of ground states. We give examples of the calculation of existence domain for the bcc and hcp lattices under the first and second nearest neighbour pair interactions. The first step is to select a basic cluster which determines the dimension of the configuration space. We select the irregular tetrahedron abcd (Figure 5.3) as our basic cluster which accounts for up to the second nearest neighbour pair interactions. The configurational variables are then the point correlation function x1 = 1 ; pair correlation functions, x2 = 1 2 x3 = 1 3 ; 3-point correlation function, x4 = 1 2 3 ; and the tetrahedron correlation function, x5 = 1 2 3 4 . ijkl Because of the constraint that 4 ≥ 0, we are led to the following consistency relations expressed in the matrix form as (Ducastelle 1991, Inden and Pitsch 1991) ⎡

⎤ 1 4 4 2 4 1 ⎢ 1 2 0 0 −2 −1 ⎥ ⎢ ⎥ ⎢ 1 0 0 −2 0 1 ⎥ ⎢ ⎥ ⎢ 1 0 −4 2 0 1 ⎥ ⎢ ⎥ ⎣ 1 −2 0 0 2 −1 ⎦ 1 −2 4 2 −4 1

⎤ 1 ⎢ x1 ⎥ ⎢ ⎥ ⎢ x2 ⎥ ⎢ ⎥≥0 ⎢ x3 ⎥ ⎢ ⎥ ⎣ x4 ⎦ x5 ⎡

If these inequalities are taken as equalities, these equations define a simplex in the five-dimensional configurational space. All the vertices correspond to existing atomic configurations. It is well known that the minimum of energy will occur for atomic configurations corresponding to the vertices of the configuration polyhedron. Its projection onto the subspace (x1 x2 x3 ) defines the configuration polyhedron (Figure 5.4). The coordinates of the vertices are given in Table 5.1. The corresponding crystallographic structures are shown in Figure 5.1 and the pertinent crystallographic data are tabulated in Table 5.2. Similarly ground state analysis can be carried out for the hcp lattice under the octahedron–tetrahedron cluster approximation (Figure 5.5), which includes up to third nearest neighbour pair interactions and gives rise to 14 correlation functions defined as follows: x1 = 1 , x2 = 1 2 , x3 = 1 6 , x4 = 3 6 , x5 = 1 6 7 , x6 = 1 3 6 , x7 = 1 2 6 , x8 = 2 3 4 , x9 = 1 2 3 4 , x10 = 1 2 3 6 , x11 = 1 2 6 7 , x12 = 2 3 6 7 , x13 = 1 2 3 6 7 , x14 = 1 2 3 6 7 8 . The ground state superstructures for the hcp lattice corresponding to the vertices of the 14-dimensional configuration polyhedron are listed in Table 5.3. A few of them are also depicted in Figure 5.5.


399

x3

x2

1 3

x1

2

4

Figure 5.4. Projection of the configuration polyhedron corresponding to the irregular tetrahedron of the bcc lattice. The vertices designated correspond, respectively, to (1) B2, (2) D03 , (3) D03 and (4) B32 ground state ordered structures.

Table 5.1. Coordinates of the vertices of the configurational polyhedron corresponding to the minimum in energy under the first and second nearest neighbour (NN) pair approximation. Configuration Pure A (A2) A3 B, AB3 D03 ) AB (B2) AB (B32) Pure B (A2)

x1

x2

x3

x4

x5

1 ± 21 0 0 −1

1 0 −1 0 1

1 0 1 −1 1

1 ± 21 0 0 −1

1 −1 1 1 1

For the fcc lattice, the octahedron–tetrahedron cluster approximation (Figure 5.6) takes into account up to the second nearest neighbour pair interactions. The configurational polyhedron corresponding to the octahedron is a simplex in the nine-dimensional space corresponding to all possible multiatom interactions. All subclusters equivalent in the fcc lattice are also equivalent in the octahedron. If we project this simplex onto the space (x1 , x2 , x3 ) corresponding to point, first and second nearest neighbour correlation functions, only the 10 vertices survive. For the tetrahedron inequalities the corresponding polyhedron is a prism. This prism

400


Table 5.2. Crystallographic data for various bcc-based ground state superstructures under the first and the second NN pair approximations. Structures bcc-based

Compositional formulae

Space group symbol (no.)

Wyckoff positions

Multiplicity

Transformed basis

A (a)

2

a = a100

A (a) B (b)

1 1

a1 = a100 a2 = a010 a3 = a001

A2

A

B2

AB

¯ (225) Im3m ¯ (221) Pm3m

B32

AB

¯ (227) Fd3m

A (a) B (b)

8 8

a1 = a200 a2 = a020 a3 = a002

D03

A3 B

¯ (225) Fm3m

A (c) A (b) B (a)

8 4 4

a1 = a200 a2 = a020 a3 = a002

The transformed basis vectors are given in terms of the vectors of the bcc lattice.

(a) Unit cell

(f) B19

(b) Octahedron (c) Tetrahedron

(g) D019

(d) CuPt

(e) A2B

(h) D0a

Figure 5.5. A few ordered structures which are the ground state superstructures under the first and the second nearest neighbour interaction approximation for the hcp lattice. Large circles are in {00n} planes and small circles are in {00n + 21 } planes. (a) Unit cell of hcp structure, (b) octahedron, (c) tetrahedron, (d) CuPt type, (e) A2 B, (f) B19, (g) D019 and (h) D0a .

cuts the octahedron polyhedron. The final polyhedron is shown in Figure 5.7. It has 15 vertices out of which only 9 vertices are really distinct. These vertices are described in Table 5.4 and the corresponding superstructures are also shown in Figure 5.8; relevant crystallographic data are listed in Table 5.5.


401

Table 5.3. Ground state structures of the hcp lattice corresponding to the vertices of the 14-dimensional configurational polyhedron. Concentration

Prototype

A AB AB AB A3 B A3 B A2 B A2 B A2 B A5 B A4 B3

Mg AuCd CuTe WC Ni3 Sn -Cu3 Ti

Designation A3 B19 Bh D019 D0a

B2 NdRh3 Si2 Zr B2 NdRh3

C49

b a c f

e

d g

Figure 5.6. The tetrahedron (abcd)–octahedron (bcdefg) (TO) cluster approximation for the fcc lattice.

5.2.4 Special point ordering A wide range of phenomena related to order–disorder and magnetic transitions can be explained using the symmetry properties of the pair potentials (Vij ). One such example is the Landau theory of continuous phase transitions. The free energy of an alloy system for different wave vectors corresponding to appropriate concentration fluctuations has been shown by Khachaturyan (1978) and de Fontaine (1979) to be a useful parameter in the study of the instabilities associated with ordering phase transitions. The roles that symmetry and wave vectors, terminating at special points of the reciprocal lattice, play can be illustrated as follows. If a symmetry element of the space group in k-space is located at a point h, the vector representing the gradient, h Vh, of an arbitrary potential energy function Vh at that point must lie along or within the symmetry element. If two or more symmetry elements intersect at the point h, one must necessarily have h Vh = 0

(5.20)

402

Phase Transformations: Titanium and Zirconium Alloys x3

1 2

x2 8

3 4 7 5 x1 6

9

Figure 5.7. Dual configurational polyhedron for the fcc lattice under the tetrahedron–octahedron cluster approximation. The vertices designated correspond, respectively, to (1) pure A, (2) L12 , (3) D022 , (4) A5 B, (5) Pt2 Mo (Immm), (6) A2 B, (7) A2 B2 , (8) L10 and (9) L11 . Table 5.4. Ground state structures of the fcc lattice corresponding to the vertices of the nine-dimensional configurational polyhedron. Configurations

x1

x2

x3

A

1 2 3 1 2 1 2 1 3

1 1 3

1 1 3

0

1

A5 B L12 D022 Pt2 Mo

0 1 9

2 3 1 9

A2 B

1 3

0

1 3

L10

0

1 3

1

A2 B2

0

1 3

1 3

L11

0

0

1


L12

D022

L10

A2B2

403

L11

A2B (Immm )

Figure 5.8. fcc-based ordered intermetallic phases which are the ground state superstructures under the first and the second NN pair approximation.

At these so-called special points, the potential energy function, Vh, represents an extremum regardless of the choice of the pair interaction energies. Thus special points play an important role in the search for lowest energy ordered structures. The special points are, however, quite insufficient for a complete description of the ground states which can be treated conveniently in the real space as demonstrated in the last section. Among the most important applications of special points is the study of the onset of short wavelength instabilities in alloys. Such instabilities may take place at a high supercooling below a first order transition temperature, frequently bearing no symmetry relation to either the low or the high temperature phases. This particular metastable ordering mechanism, known as spinodal ordering, has been observed in several alloy systems (e.g. Ni–Mo alloys). The points which differ by a lattice vector of a reciprocal lattice are considered equivalent. In the case of simple structures with a single atom per unit cell, it is sufficient that two symmetry elements intersect at special points. These special points are listed in crystallographic tables. They are always located at the surface of the Brillouin zone. The “star” of a special point vector k is obtained by applying all the rotations and rotation-inversion of the space group on the vector k. All these vectors of a star are also considered equivalent.

404


Table 5.5. Crystallographic data for various fcc-based ground state superstructures under the first and the second NN pair approximation. Structures bcc-based

Compositional formulae

Space group symbol (no.)

Wyckoff positions

Multiplicity

Transformed basis

A1

A

¯ (225) Fm3m

A (a)

4

a = a100

L10

AB

P4/mmm (123)

A (a) B (d)

1 1

a1 = 21 a110 a2 = 21 a110 c = a001

L11

AB

¯ (166) R3m

A (a) B (b)

1 1

a1 = 21 a110 a2 = 21 a101 c = a222

A2 B2

A2 B2

I41 /amd (141)

A (a) B (b)

4 4

a1 = a010 a2 = a001 c = a200

Pt2 Mo

A2 B

Immm (71)

A (i) B (a)

4 2

a = 21 a110 b = a001 c = 21 a330

A2 B

A2 B

B2/m (12)

A (i) A (c) B (a)

4 2 2

L12

A3 B

¯ (221) Pm3m

a = 21 a552 b = a220 c = 21 a110

A (c) B (a)

3 1

a = a100

D022

A3 B

I4/mmm (139)

A (d) A (b) B (a)

4 2 2

a1 = a010 a2 = a001 c = a200

Ni4 Mo type

A4 B

I4/m (87)

A(h) B(a)

8 2

a1 = 21 a310 a2 = a130 c = 002

The transformed basis vectors are given in terms of the vectors of the fcc lattice.

5.2.4.1 BCC special points The special points of the bcc structure are located at the points ", H, P and N of the Brillouin zone (Figure 5.9). The corresponding stars are given in Table 5.6 (Ducastelle 1991). Here we have a single structure per star. The domain of stability in the case of nearest neighbour and next nearest neighbour interactions is shown in Figure 5.10. The star 21 21 0 does not appear which is understood by noting that the corresponding structure has not been obtained as a possible ground state which requires up to third NN interactions. The only ground state that does not appear is the D03 structure which can be constructed by a superposition of 100 and 21 21 21 waves.


405

kZ

H Γ

kY

N

H

kX

Figure 5.9. The Brillouin zone for the bcc lattice where various special points have been marked. Table 5.6. The special points and stars of the bcc structure. k-Vector star

Members

000

000

"

100

1 1 1

100 1 1 11 1 1

H

B2

P

B32

N

AB

2 2 2

1 1 0 2 2

2 2 2

Brillouin zone points

2 2 2

1 1 1 1 1 1 0 202 02 2 2 2 1 1 1 1 1 1 0 202 02 2 2 2

Ordering structure

V2 〈

111 〉 222 (a)

(b)

V1 〈000〉

〈100〉

Figure 5.10. The domains of stability of the stars for the bcc lattice in the first (V1 ) and the second (V2 ) nearest neighbour interaction plane. The lines designated (a) and (b) correspond, respectively, to (a) 2V1 − 3V2 = 0 and (b) 2V1 + 3V2 = 0.

406


5.2.4.2 HCP special points The analysis of the hcp structure is much more complex as it has two atoms per unit cell and the space group of the structure (P63 /mmc) is not symmorphic. There are six special points for the hcp lattice, viz., " (000), M ( 21 00), K ( 13 31 0), L ( 21 0 21 ), H ( 13 31 21 ) and A (00 21 ). Using symmetry arguments, it can be shown (Ducastelle 1991) that some of these special points are irrelevant for the hcp structure. The three relevant special points are ", M and H (Table 5.7). At the points " and M, there are two extremes and we must combine the concentration waves in the two sublattices in two different ways. These combinations are obtained from the eigenvectors of the potential energy matrix V k. The result is that at " and M, we have simple modes which may be called acoustic and optical modes as in the theory of lattice vibrations. If we let the origin of sublattice 1 be at (000) then the origin of sublattice 2 is at 23 13 21 . The stars are tabulated in Table 5.7 and the structures corresponding to the pure modes are described below (Figure 5.11). Table 5.7. Important special points and stars of the hcp structure. k-Vector star

Members

000

1 00 2 1 1 1

000 1 1 1 1 00 0 2 0 2 2 0 2 1 1 11 1 1

332

33 2

Brillouin zone points " M H

332

V2

(b)

〈000〉 ; 〈 1 00〉 2 II I

(a)

〈 〈000〉

11 0〉 44

V1

I 1 〈 00〉 2 II

Figure 5.11. The domains of stability of the stars for the hcp lattice in the first (V1 ) and the second (V2 ) nearest neighbour interaction plane. The lines designated correspond, respectively, to (a) V1 − 2V2 = 0 and (b) V1 + V2 = 0. In the hatched region, Vh is minimum at 41 41 0 , which is not a special point.


407

Star 000 : The acoustic mode 000 I corresponds to the segregation process and the optical mode 000 II corresponds to an alternate stacking of A and B triangular planes (CuPt-type structure) Star 21 00 : Here we regard 21 00 I and 21 00 II as modes corresponding to the CuPt type and MgCd (B19) type structures. Also we obtain the D019 structure by considering the acoustic combination of k1 = 21 00, k2 = 0 21 0 and k3 = 21 21 0. There is no structure corresponding to the optical mode. Star 13 31 21 : There is no structure associated with this star. In the case of nearest neighbour interactions, V (k) is never a minimum at H. When the second nearest neighbour interaction is introduced, the absolute minimum of V (k) in the range 0 < V2 < 21 V1 occurs at the point [ 41 41 0 ] that does not correspond to a special point but rather is a “non-special” point. 5.2.4.3 FCC special points The special points of the fcc structure are located at the points ", X, W and L of the Brillouin zone (Figure 5.12). The corresponding stars are given in Table 5.8 (Ducastelle 1991). The domain of stability in the case of nearest neighbour and next nearest neighbour interactions is shown in Figure 5.13. 5.2.5 Concomitant clustering and ordering The thermodynamic analysis of concomitant clustering and ordering was presented by Kulkarni et al. (1985) and Soffa and Laughlin (1988, 1989). The latter theoretically studied various barrierless reaction mechanism for the transition

kZ

L Γ

X kX

kY W

Figure 5.12. The Brillouin zone for the fcc lattice where various special points have been marked.

408

Phase Transformations: Titanium and Zirconium Alloys Table 5.8. The special points and stars of the fcc structure. k-Vector star

Members

000

000

"

100

1 120

100 010 001 1 1 1 1 2 0 2 01 01 2 1 21 0 21 01 01 21 0

X

L12 , L10

W

A2 B2

L

L11

1 1 1 2 2 2

1 1 1 2 2 2

1 1 1 2 2 2

1 1 1 2 2 2

1 1 1 2 2 2

Brillouin zone points

Ordering structure

V2 〈

111 〉 222 (a)

(b) 1 〈1 0〉 2 〈000〉

V1 〈100〉

Figure 5.13. The domains of stability of the stars for the fcc lattice in the first (V1 ) and the second (V2 ) nearest neighbour interaction plane. The lines designated (a) and (b) correspond, respectively, to (a) V1 − 2V2 = 0 and (b) V1 + V2 = 0.

“fcc disordered solid solution → ordered state having L12 structure” by employing the graphical thermodynamics approach and the SCW model. Soffa and Laughlin have expounded the idea of ordering and clustering reactions in constructing the schematic free energy versus composition plots which illustrate different situations. The stability/instability of a solution with respect to change of some parameter, e.g. an appropriate order parameter or the site occupancy, is determined by the second derivative of the free energy (F ) with respect to that variable. A solution is considered stable, critical or unstable according to whether F > 0, F = 0 or F < 0, respectively. The equation F = 0 describes the point at


409

which an instability develops in the solution. While the presence of an instability can be determined by examining the sign of the second derivative of free energy with respect to appropriate order parameter, the equilibrium phase boundaries are given by the principle of common-tangents on the free energy–concentration curves for the ordered and the disordered phases. The instability lines on the temperature (T )–concentration (c) plane are defined as follows: Ti− : the line below which the solid solution is unstable with respect to congruent ordering Ti+ : the line above which the ordered solid solution becomes unstable with respect to spontaneous congruent disordering Tcs : the line below which spinodal clustering instability develops only after the system undergoes ordering to a certain extent (conditional spinodal) Ts : the line below which the disordered phase becomes unstable with respect to spinodal clustering. The two equilibrium phase boundaries (Teq s) define the ordered, disordered and the two-phase regions. The four instability lines Ti− , Ts , Ti+ and Tcs are categorized in terms of which phase forms the instability: the disordered () or the ordered () phase. Accordingly, Ti− and Ts are the instabilities of the disordered phase, being the points at which the disordered phase becomes unstable with respect to ordering and clustering, respectively. The other instabilities, T+i and Tcs , are the points at which the ordered phase becomes unstable with respect to spontaneous disordering and clustering, respectively. The computation of the disordered phase instabilities is straightforward and can be done exactly, while the ordered phase instabilities are somewhat more complex and can be determined from the graphical thermodynamic approach. The relative positions of these instability lines, as schematically illustrated in the phase diagram (Figure 5.14), identify the conditions under which different transformation sequences are possible from thermodynamic considerations. In the cases where barrierless mechanisms are operative, the phase transformation mechanism will be determined by which of the instability lines have been crossed on quenching. Much research has been dedicated to the study of complex instability reactions that result in simultaneous ordering and phase separation in alloys. Laughlin and Soffa compared these results (Soffa and Laughlin 1989) to phenomena observed in several systems involving → L12 phase transition. It is possible for a miscibility gap to develop in the L12 phase at a temperature where the curvature of the free energy is negative. It is equally possible to have a miscibility gap develop in the disordered phase by the development of a negative

410


Ti + B′

B

Ti –

C

Ts

Temperature

Temperature

A D Ts E Ti –

Tcs

Composition

Composition

(a)

(b)

Figure 5.14. The instability diagrams which schematically illustrate the relative positions of all the four instability lines (Ti− , Ti+ , Tcs and Ts ) superimposed on the phase diagram. The two equilibrium phase boundaries are drawn with thick lines. (a) shows ordering instability line (Ti− ) at higher temperatures compared to clustering instability line (Ts ) while in (b) Ts is above Ti− . These lines segment the phase diagram into several domains in which different reaction mechanisms are operative. Domain A: → + (nucleation and growth). Domain B: → spinodal ordering → + . Domain C: → simultaneous ordering and clustering. Domain B : → spinodal ordering → spinodal clustering. Domain D: → spinodal clustering → spinodal ordering within solute-enriched regions → + . Domain E: simultaneous ordering and clustering.

curvature along that branch of the free energy. Depending on the temperature at which these develop, a variety of phase diagrams can be produced. They gave results for both first order and second order ordering reaction cases. In all, five different phase diagrams (shown in Figure 5.15; Table 5.9), along with the instability lines that would be obtained in each case are: (a) the classical phase separation case, (b) the first-order ordering case, (c) a monotectoid case, in which a miscibility gap develops in the ordered phase, (d) one in which a miscibility gap develops in the disordered phase that is metastable with respect to the ordering reaction and (e) a syntectoid case, in which a miscibility gap develops in the disordered phase at high temperatures and an ordering reaction occurs at low temperatures. An alloy of composition co is in a single phase disordered region (Figure 5.15(a)) at higher temperatures (above the miscibility gap maximum) and within a


C1 A

X

α

α2

Coherent spinodal α1 + α2 C0

II + III

Ti

+

α

β

Ti

–

V IV

C2

Composition

Ti

+

Composition

Composition

(a)

II + III – Ti Vβ

IV

Ax By B

Ax By

I

I

Temperature

α1

α Miscibility gap I Chemical spinodal

Temperature

Temperature

α0

411

(c)

(b) T cs

X

I

Temperature

II + III

Ti

+

β

IV + V VI + VII

–

VII + IX

Ti

+

β

Ti

–

VIII VI

Ax By

Composition

(d)

Ti

Temperature

α

I

VI

Ax By Composition

(e)

Figure 5.15. The phase diagrams produced by systematically varying the curvatures of the free energy curves for the ordered and disordered phases. The phase diagram (c) is not observed in the SCW model. (a) A simple phase separation case, (b) a simple ordering diagram, (c) monotectoid diagram, (d) ordering with metastable miscibility gap diagram and (e) a syntectoid case. The stability regions have also been shown (see Table 5.9, after Soffa and Laughlin 1989).

two-phase ordered region at lower temperatures. The two phases differ from one another only in composition but have identical crystal structure, i.e. isostructural decomposition, whereas in the cross-hatched region in Figure 15.5(b), the homogeneously ordered phase is unstable with respect to phase separation. The Ti+ is the instability line with respect to disordering upon heating. In Figure 15.5(c), the cross-hatched region is the region of thermodynamic instability with respect to phase separation. The Laughlin and Soffa approach has the advantage of simplicity and generality; it does not rely on the assumption of any particular model. It has the disadvantage that the thermodynamic consistency is not built-in, i.e. there are consistency rules for construction of free energy curves that have not been examined.

412

Phase Transformations: Titanium and Zirconium Alloys Table 5.9. The characteristic barrierless reactions operative in different stability regions as predicted by the SCW model (Simmons 1992). Region

Reaction

Region

Reaction

I III

No reaction No reaction

II IV

V

→

VI

VII

→

VIII

IX

→ + →

X (T > Tconsolute ) (T < Tconsolute )

No reaction → → + → → → + → → + → → + → + →

≡ disordered phase and ≡ ordered phase.

5.2.6 A case study: Ti–Al system As an illustration of what has been discussed so far, we will present here a summary of the results obtained on the Ti–Al alloy system using the first principles configurational thermodynamic approach by Asta et al. (1992, 1993). The Ti–Al system is an incoherent system as the different stable/metastable ordered phases that appear across the full composition range are the superstructures of either the hcp (corresponding to -Ti) or the fcc (corresponding to Al) lattice. Prior to the work of Asta et al., several ab initio electronic structure calculations had been carried out on the Ti–Al system (Fu 1990, Hong et al. 1991) to study the energetics as well as the structural and mechanical properties of stable and metastable stoichiometric compounds at 0 K. But the effect of variation of temperature and composition on these properties was not studied. Asta et al. (1992) have used the local density based full potential linear muffin tin orbital (FP-LMTO) method to calculate the ground state cohesive properties of all the stable and metastable phases under the tetrahedron–octahedron cluster approximation for both fcc and hcp lattices (Table 5.10). The ECI were calculated using the Connolly–Williams inversion method and the CVM free energies were used to determine the phase stabilities as functions of temperature, composition and order parameter to arrive at the Ti–Al phase diagram depicting the stability regimes of various stable and metastable phases and the pertinent transition temperatures. Their results showed that of all the structures considered, only hcp-Ti, fcc-Al, D019 -Ti3 Al, L10 -TiAl and D022 -TiAl3 were energetically stable in agreement with the experimental results. The remaining phases were found to be metastable. Of


413

Table 5.10. Cohesive properties of some of the intermetallics of Ti–Al system. System

Lattice parameter (a, (c/a)) (Å)

Bulk modulus (GPa)

Al

3.985 4.046 2.864 2.951 5.649 5.761 3.935 4.005 3.790 3.848

84 82 106 105 126 120–145 128 160–176 118 —

Ti Ti3 Al TiAl TiAl3

(1.000) (1.000) (1.617) (1.588) (0.809) (0.807) (1.012) (1.016) (2.240) (2.234)

Formation energy (kJ/mol) 0.000 0.000 −287 −253 −420 −3650 −419 −307 to −374

In the first row of each column are the calculated results as obtained by Asta et al. using the FP-LMTO method; in the second row are the experimental results.

these metastable phases, L12 -TiAl3 was found to be in close competition with D022 -TiAl3 phase. Further, the hcp disordered alloys were noted to be stable as compared to fcc disordered solid solutions up to 50 at.% Al. The Al-rich and equiatomic compounds were shown to undergo large structural relaxations even though the size mismatch between the atomic constituents was small. The calculated CVM phase diagrams for the fcc- and hcp-based Ti–Al alloys showed that ordered fcc-based alloy phases are more stable than hcp-based alloy phases with respect to corresponding disordered solid solution. Consequently there is a stronger tendency for atomic ordering in fcc-based alloys which was attributed to the differences in the electronic structures between fcc- and hcp-based alloy phases. At Al-rich compositions, the L10 and the D022 phases were found to be strongly ordered to well above their experimentally observed melting temperatures. Moreover the phase field of D022 -structured TiAl3 was predicted to be much narrower than that for L10 -TiAl which is in agreement with the experimental phase diagram. The theoretical transition temperature for the D019 -Ti3 Al phase was about 600 C too high; this discrepancy was attributed to the neglect of contributions from vibrational and electronic excitations to the free energy. The phase relationships in the Ti–Al system have been subjected to many modifications in the recent past, partially because of differences in the impurity levels in the alloys studied and in the interpretation of the observed microstructures. There have been major discrepancies concerning the extension of the (-Ti) field with respect to temperature. These modifications are also reflected in the various calculations published in the literature. Kauffmann and Nesor (1978) used very simple descriptions in their calculations: the disordered solution phases were

414


described as quasiregular solutions and the intermetallic compounds Ti3 Al, TiAl and TiAl3 were assumed to be stoichiometric. Within the limitations of the model descriptions used, the general features of the then accepted phase diagram could be reproduced. Murray (1987) used the subregular solution model to describe the disordered phases and assumed TiAl to be stoichiometric. Gros et al. (1988) calculated the Ti-rich part of the system. The liquid and (-Ti) phases were described as quasiregular solutions and (-Ti) and Ti3 Al were described in terms of the sublattice model as one single phase undergoing an order–disorder transition. The calculated partial Gibbs free energies of Al and Ti in (-Ti) agreed within 5–12% of the experimental values. Within the constraints used, reasonable agreement was obtained for the (-Ti) and Ti3 Al phase boundaries. More complete calculations of the Ti–Al system, taking into account the entire phase diagram and the ranges of homogeneity of the ordered intermetallic phases, were carried out later by Murray (1988). He described the disordered solution phases as quasiregular solutions, used the Bragg–Williams approximation for the ordered Ti3 Al and TiAl compounds and assumed TiAl3 to be stoichiometric. In order to improve the agreement between the calculated and experimental phase diagrams, Murray found it necessary to relax the conditions given by the Bragg– Williams approximation for the Gibbs free energy for these compounds. Later, Hsieh et al. (1987) and Lin et al. (1988) included the TiAl2 and Ti3 Al7 phases in the calculated phase diagram by using a quasisubregular solution model. Recently, a thorough thermodynamic assessment of the Ti–Al system has been undertaken by Zhang et al. (1997) in which three different analytical descriptions have been used to describe the three different types of phases occurring in the system: the stoichiometric compounds, the disordered solution phases and the ordered intermetallic compounds which have homogeneity ranges. A least square technique has been used to optimize the thermodynamic quantities pertinent to the analytical description, using the experimental data available in the literature. We will give here a brief account of the approach used by Zhang et al. (1997) for the assessment of the Ti–Al phase diagram in order to illustrate the semi-empirical methodology adopted. For an accurate description of these phases, the Gibbs free energy (G) must be expressed as an analytical function of the thermodynamical variables of interest, viz., composition, temperature and pressure. The Bragg–Williams model is inadequate to account for the phase reactions/transitions in this system because of its inability to account for local correlations or short-range ordering effects. Zhang et al., therefore, have used different models for the ordered and disordered phases. They have considered nine phases in their calculations: the disordered solution phases, liquid, -Ti, -Ti and Al; and the ordered intermetallic compounds Ti3 Al, TiAl, TiAl2 , Ti2 Al5 and TiAl3 . The TiAl2 and Ti2 Al5 phases have been assumed


415

to be stoichiometric, while the wide ranges of homogeneity exhibited by Ti3 Al and TiAl have been taken into account in the analytical model. The free energies corresponding to the stoichiometric compounds TiAl2 and Ti2 Al5 are described by o o GoTi + xAl GoAl + Gf G = xTi

(5.21)

where xio are the mole fractions of component i and Goi are the respective reference states. The term Gf represents the Gibbs free energy of formation. These G parameters are either constant or are linear functions of temperature. The disordered solution phases, liquid, -Ti, -Ti and Al are described as random mixtures of Ti and Al: xi ln xi + xTi xAl Go + G1 xTi − xAl (5.22) G = xTi GoTi + xAl GoAl + RT i

where Go and G1 are the coefficients of the terms of the excess Gibbs energy. This model is called the quasisubregular solution model. The ordered intermetallic compounds Ti3 Al, TiAl and TiAl3 exhibit an appreciable range of homogeneity and are described by the sublattice or Wagner–Schottky model. These compounds are considered to consist of two sublattices which are occupied by Ti and Al atoms only in the perfectly ordered state. The partially ordered regions are described by assuming the formation of substitutional atoms on each sublattice. The analytical description is 1 2 1 o o ni ln n1i + RT ni ln n2i − RT Ni ln Ni1 G = xTi GTi + xAl GAl + RT i

−

i

Ni2 ln Ni2 + Gf +

i

i

n2i G2i + n1Ti n1Al G1o + G11 n1Ti − n1Al

i

+ n2Ti n2Al G2o + G21 n2Ti − n2Al + n2Ti n1Al G12 Here we have xi = n1i + n2i and N k = nkTi + nkAl where nki are the mole fractions of component i on sublattice k and N k are the site fractions of sublattice k. Gf is the free energy of formation of the perfectly ordered phase at the stoichiometric composition. Gko are the coefficients of polynomial interaction terms between atoms on the same sublattice and G12 is that between substitutional atoms on the different sublattices. The quantities nki are calculated by minimizing the Gibbs free energy for given xi . The optimization and calculation programmes developed by Lukas et al. (1982) have been used for these calculations of the Ti–Al system.

416

5.3


TRANSFORMATIONS IN TI3 AL-BASED ALLOYS

5.3.1 → D019 ordering The 2 -phase based on the stoichiometry Ti3 Al has a D019 (hP8) structure and P63 /mmc symmetry. The structure is characterized by an atomic arrangement on the close packed (0001) planes (Figure 1.20) which ensures that Al atoms have only Ti atoms as their first near neighbour. A hexagonal stacking ABABAB of such planes produces the D019 structure. An examination of the symmetry rules for a second order ordering process in the context of → D019 transformation reveals that the first Landau–Lifshitz condition (symmetry elements of the product phase having a subset of that of the parent) is fulfilled. This means that replacement of atoms in an -lattice can produce a D019 structure. The D019 structure can be formed by introduction and amplification of a concentra tion wave with wave vector star of k 21 00 . This wave vector terminates at the special point of the reciprocal space fulfilling the second Landau–Lifshitz condition. The third condition, however, is not satisfied as the sum of three variants of the wave vector star (k1 + k2 + k3 ) results in a reciprocal lattice vector of the parent hcp structure. Therefore → D019 transformation is not a candidate for a second order ordering process. With a high degree of supercooling, at T< Ti , Ti being the instability, a short wave length concentration wave with k= 21 00 , can amplify at least to a limited extent to generate a partially ordered D019 structure. The formation of the Ti3 X precipitates having the D019 structure is seen in a number of binary and ternary alloy systems such as Ti–Al, Ti–Ga, Ti–Al–Ga, Ti–Al–Sn, Ti–Al–In. It is important to note that the mismatch between the lattice dimensions between the parent and the ordered Ti3 X precipitates are different in different alloy systems and correspondingly the morphology of precipitates varies from one system to the other. The extent of mismatch between the matrix and Ti3 X D019 precipitates in different systems are listed in Table 5.11. Table 5.11. Matrix–precipitate mismatch along a and c directions for Ti–Al alloys. Alloy system

Ti–Al Ti–Al–Ga Ti–Al–Sn Ti–Al–In

Matrix–precipitate mismatch Along a (%)

Along c (%)

0.83 0.99 0.20 0.58

0.35 0.51 0.15 0.21


417

5.3.2 Phase transformations in 2 -Ti3 Al-based systems The development of engineering alloys based on Ti3 Al (2 ) was initiated with the primary objective of ductilizing the brittle binary Ti3 Al intermetallic by the introduction of -stabilizing elements, particularly Nb. As mentioned earlier, Ti3 Albased alloys can be classified into different groups on the basis of the extent of additions of -stabilizers. The relative stabilities of three phases, namely, the 2 (D019 ), o (B2) and O (orthorhombic) phases, play a key role in determining the phase equilibria in these alloys. The thermal history, however, remains as the other major factor in controlling the mode and the sequence of phase transformations and the microstructure developed. The literature on Ti3 Al-based alloys is overwhelmingly dominated by that on ternary Ti–Al–Nb alloys. The trends in the phase transformations in Ti3 Al-based alloys are, therefore, discussed in this chapter with reference to the ternary Ti–Al–Nb system. Figure 5.16 shows (a) pseudobinary phase diagram (Ti–27.5% Al with increasing Nb) and (b) the isothermal section at 1173 K of the ternary phase diagram of the Ti–Al–Nb system (Banerjee and Rowe 1993). Phase fields in which a single phase, 2 , o or O, exists are shaded and two- and three-phase fields are marked. The composition ranges associated with the three classes of alloys, containing different levels of -stabilizers, which have been studied in detail, are indicated by circles, inscribed with the legends I, II and III. This 1173 K isotherm of the ternary phase diagram suggests that alloys belonging to class I consist of a mixture of the 2 - and o -phases while those belonging to class II made up of a mixture of the o - and O-phases. The class III alloys of intermediate compositions exhibit a three-phase (2 + o + O) microstructure.

30

Nb

10

βo + O II

50Ti Al 20

20

Al

O + βo O

40

1000

Al

O

10

βo

25

α2 + βo α2

Ti–27.5Al

60Ti

O + βo + α2

30

1100

900

Nb

III

βo (B2)

Al

α + βo α + α2

Nb

α

20

1200

Temperature (°C)

I βo + α2

α2

(b) 30

Atomic % Nb (a)

Figure 5.16. (a) Pseudobinary phase diagram (Ti–27.5% Al with increasing Nb) and (b) the isothermal section at 1173 K of the ternary phase diagram of the Ti–Al–Nb system.

418


Since the compositions of all the three classes of alloys fall nearly along the line of 25 at.% Al, a pseudobinary section along the Ti3 Al–Nb line (Figure 5.16(a)) can conveniently be used to show the possible phase reactions in these alloys. A comparison can be drawn between this pseudobinary phase diagram and a binary phase diagram of Ti alloyed with a -stabilizing element (Figure 6.1). The similarity between them essentially arises due to their -isomorphous nature which restricts the composition range of the /2 phase field. Another point of similarity is reflected in the tendency for the formation of an orthorhombic structure at relatively high levels of -stabilizer addition. In the case of the binary Ti–X (X being a -stabilizer like V, Mo, Nb) system the orthorhombic phase ( , orthorhombic martensite) is an extension of the hcp -martensite. In comparison, the orthorhombic O-phase in the Ti3 Al–Nb phase diagram can be considered as an extension of the Ti3 Al (D019 ) phase with an orthorhombic distortion. This argument will be further elaborated when the crystallography of these phases is considered. The essential difference between the binary Ti–X and the pseudobinary Ti3 Al–Nb phase diagrams is due to the fact that there is a strong tendency towards chemical ordering in the latter case. In Ti3 Al-based alloys with low Nb contents, the terminal solid solution experiences a strong tendency to order into the D019 structure while alloys more enriched with Nb are under the influence of a strong tendency towards B2 ordering at elevated temperatures and O-phase ordering at lower temperatures. The presence of these ternary ordering tendencies is responsible for the introduction of additional phase reactions in this system. Having discussed the general tendencies related to phase stabilities in Ti3 Albased alloys, let us examine the experimental observations made on this system. Banerjee (1994a,b), in his excellent review, has constructed a diagram in which experimental observations on phase transformation processes in this system have been summarized. Figure 5.17 is essentially a reproduction of the same diagram in which the data reported in a number of research papers have been assimilated (Banerjee et al. 1988; Kestner-Weykamp et al. 1989, 1990; Bendersky et al. 1991; Kattner and Boettinger 1992; Muraleedharan et al. 1992a,b; Rowe et al. 1992). At low Nb levels ( and < 112 > directions is noticed (Figure 5.18(c)). These instabilities have been ¯ < 110 ¯ > and attributed to transverse displacement waves which are of the 110 ¯ 112 < 111 > types. Strychor et al. (1988) have pointed out that the former is associated with a precursor for the → martensitic transformation while the latter originates from the tendency for the occurrence of the → # transition. A tweed contrast observed in the quenched-in o (ordered ) phase (Figure 5.18(d)) is consistent with the presence of transverse displacive waves and this tendency is noticed over the entire range of Nb concentration from 7.5 to 25 at.%. A relatively slow cooling from the -phase induces diffusional phase transformations in alloys containing 7.5–25 at.% Nb, leading to a mixture of 2 + o ,

420


(a)

(c)

(b)

(d)

Figure 5.18. (a) Fine D019 domains within the martensite laths in quenched alloys containing up to 7.5 at.% Nb, (b) quenched-in -phase shows a domain structure resulting from the bcc to B2 ordering, (c) in addition to the presence of B2 superlattice reflections, extensive streaking along < 110 > and < 112 > directions is shown and (d) a tweed contrast as observed in the quenched-in o (ordered ) phase (after Strychor et al. 1988).

2 + o + O or O + o phases. The parent o -phase undergoes a decomposition into either 2 + o or O + o phase mixtures which is followed by the subsidiary phase reaction, 2 → 2 + O or 2 + o → O. The precipitation of the 2 -or the O-phase results in a lath, lamellar or mosaic morphology, depending on the alloy composition and the cooling rate. The Burgers orientation relationship between the hexagonal or the orthorhombic product and the parent cubic phase is invariably maintained. The volume fraction of the primary 2 -phase (which subsequently transforms to the O-phase) decreases with increasing Nb content and remains confined to grain boundaries as the alloy composition approaches 25 at.% Nb. The morphology of the product, at the level of light microscopy, bears a significant


421

resemblance to that observed in conventional + Ti alloys slowly cooled from the -phase field. An increase in the cooling rate results in a refinement in the size of the packet (which is constituted of a group of laths stacked in a parallel array) and in the formation of a fine basket weave morphology. As mentioned earlier, quenching from the -phase field yields an ordered martensitic structure (2 ) in alloys containing up to 7.5 at.% Nb and leads to the retention of the o -phase in alloys containing 7.5–25 at.% Nb. Tempering of the martensite results in a rapid growth of the B2 domains (Sastry and Lipsitt 1977) and causes -phase precipitation and recrystallization to a limited extent (Martin et al. 1980). The retained o -phase, on ageing, decomposes to #-related structures at temperatures below about 770 K. The transformation from the B2 structure to a variety of ordered #-structures has been discussed in detail in Chapter 6. Ageing of the quenched-in o -phase in the composition range of 7.5–11 at.% Nb induces a massive transformation in the 2 -phase. Alloys with higher Nb contents (11– 25 at.%) undergo a o → O transformation on ageing; this can occur either by a massive transformation producing equiaxed O-phase grains or by a Widmanstatten precipitation of O-phase plates which exhibit a martensite-like substructure. These two decomposition modes can operate in parallel or sequentially with the equiaxed O-phase grains finally consuming all the O-plates (Bendersky et al. 1991; Muraleedharan et al. 1992a,b). The O-phase subsequently decomposes into the equilibrium 2 - and/or the o -phases. 5.3.3 Structural relationships The phases which are present in Ti3 Al–Nb alloys (Nb content up to 30 at.%) are all based on two basic structures: bcc at high temperatures and hcp at low temperatures. While the bcc and hcp structures are related by the well-known Burgers lattice correspondence (discussed in detail in Chapter 4), the hcp → 2 (D019 ) and the A2 → o (B2) transformations involve replacive ordering in which the crystallographic axes of the product remain parallel to those of the parent. A structural description of these ordering processes has been given in earlier sections. With increasing Nb content the hexagonal symmetry of the 2 -phase gets distorted to orthorhombic symmetry (O-phase) in a manner quite similar to the structural change associated with the transition from hexagonal ( ) to orthorhombic ( ) martensite. The lattice correspondence between the and the (Bagariatskii) is essentially the same as the Burgers correspondence as can be seen by changing from the hexagonal to the orthorhombic axes system. The martensite phase is disordered and has a Cmcm space group with four atoms per unit cell. The O-phase in the ternary Ti–Al–Nb system corresponds to the stoichiometry of Ti2 AlNb where Nb atoms occupy distinctive sublattices (Banerjee et al. 1988, Mozer et al. 1990). The A2 BC structure shown in Figure 5.19 is

422


Figure 5.19. The A2 BC structure, a prototype of the structure of the Ti2 AlNb phase (O-phase), is closely related to the D019 structure.

the prototype of the Ti2 AlNb phase (O-phase) and is closely related to the D019 structure. Crystallographic data pertaining to the ordered A2 BC structure are given in Table 5.12. In this structure, the atoms can occupy a range of positions without destroying the symmetry and the stacking sequence of the O-phase. The ranges Table 5.12. Atomic positions of Ti, Al and Nb atoms in the orthorhombic O-phase. Atom

Positions (x y z)

Ti

0.2310, 0.7690, 0.2310, 0.7690, 0.7310, 0.7310, 0.2690, 0.2690,

0.9041, 0.0959, 0.0959, 0.9041, 0.4041, 0.5959, 0.4041, 0.5959,

0.2500 0.7500 0.7500 0.2500 0.2500 0.7500 0.2500 0.7500

Al

0.0000, 0.0000, 0.5000, 0.5000,

0.1630, 0.8367, 0.6633, 0.3367,

0.2500 0.7500 0.2500 0.7500

Nb

0.0000, 0.0000, 0.5000, 0.5000,

0.6357, 0.3643, 0.1357, 0.8643,

0.2500 0.7500 0.2500 0.7500


423

of atomic coordinates for each equivalent position are also given in the table. The atomic sites of the Ti2 AlNb unit cell proposed by Mozer et al. (1990) lie within these ranges and this structure √ can be considered to be a special case of the A2 BC structure with b/a less than 3 and c/a less than the ideal value. The similarity between the D019 structure and the O-phase structure can be seen by comparing the basal plane projection of the former with the (001) plane projection of the latter (Figure 5.20). Comparing the first, second, third and fourth near neighbour bond lengths in the D019 structure with a non-ideal c/a ratio (close to that of Ti3 Al) and the corresponding bond lengths in the Ti2 AlNb phase, Singh et al. (1994) have shown that these structures have a close similarity. It may be noted that the O-phase structure involves further ternary ordering of the 2 -phase with Ti, Al and Nb atoms predominantly occupying three different types of sites (Banerjee et al. 1988, Mozer et al. 1990). In view of the structural relations associated with these phases, the relevant structural changes can be described in terms of the following:

C

01 (001)

02 (001)

A

C

A 0.970 nm

α2 [0001]

A

B

B

C

B

[2110]

1.001 nm

[010]

[0110]

(1) distortion of {110} planes and changes in their interplanar spacings (2) shuffles, or relative displacements of neighbouring {110} planes, and

[100] Ti2AlNb 0.578 nm Atom Layer A

0.595 nm Al

Ti

Nb

Angle ABC

α2

01

02

60° 63° 65°

Layer B. c /2 above

Figure 5.20. A comparison between the D019 and the O-phase structures as elucidated by the basal plane projection of the former with the (001) plane projection of the latter.

424


(3) chemical ordering that changes the occupancies of Ti, Al and Nb atoms among the lattice sites. The main advantage of describing the phase transitions in this system using a common framework is that a single thermodynamic potential can be identified as a continuous function of a set of order parameters which describe the three types of structural change. Bendersky et al. (1994) have considered the group/subgroup symmetry relationship with respect to the transformations in this system and have identified the possible sequence of crystallographic transitions. 5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63 /mmc) and ordered orthorhombic (Cmcm) phases A continuous phase transition of first or higher order requires that the symmetry of the product phase is a subgroup of that of the parent phase. This is, in fact, the Landau–Lifshitz rule I for a transition to qualify as being of second order (refer to Chapter 2). Usually the low-temperature phase has a lower symmetry than the high-temperature phase, and a transition involving a lowering of symmetry is known as ordering while one associated with an increase in symmetry corresponds to disordering. The approach pertinent to determining the possible sequence of transformations is based on successive stages of symmetry reduction. The necessary information regarding symmetry relations is contained in the space group tables of the International Tables for Crystallography (1991). On the basis of this information, the individual steps of a transition can be predicted by considering the maximal subgroup relation between the parent and the product phases in any given step. A subgroup H of the space group G is called a maximal subgroup of G if there is no subgroup L of G such that H is a subgroup of L. This approach helps in the identification of all symmetry reduction steps though it is not necessary that each of these steps must really occur in a transformation sequence. There are no apparent subgroup relations between structures with cubic and hexagonal symmetries, G1 and G2 , respectively, due to the presence of noncoinciding threefold cubic and sixfold [0001] hexagonal symmetry axes. A connection between these symmetries can be arrived at by introducing an intermediate structure with space group Gt which is either a supergroup of both the structures, G1 and G2 , or a subgroup of both the structures. A supergroup, constituting a group union of G1 and G2 , does not exist as both the bcc and hcp structures exhibit very high symmetry. However, a subgroup Gt can be found as the intersection group of G1 and G2 In particular, considering the disordered bcc and hcp structures with Im3m and P63 /mmc space groups, respectively, and taking ¯ [1120] ¯ , into account the Burgers orientation relationship, (110) 0001 ; [111]


425

between them, the intersection group Gt is found to be the orthorhombic space group Cmcm, with its c-axis parallel to the [110] direction. The Cmcm space group, with an appropriate choice of Wyckoff sites, can represent a structure which is close to hcp but differs from it in symmetry and in the relative positions of atoms on the basal plane (Figure 5.21). The Cmcm structure can also be considered as the bcc structure distorted by relative shifts of the (110) planes. Bendersky et al. (1991) have proposed sequences of transitions that connect the higher symmetry cubic and hexagonal space groups to the lower symmetry orthorhombic space group (Figure 5.22). These sequences include all known equilibrium phases observed in Ti3 Al–Nb alloys with Nb contents of less than 30 at.%. The sequences shown in Figure 5.22 show two branches, one starting from the disordered bcc structure and the other from the ordered B2 structure. The directions of symmetry reduction are indicated by the arrows, and the indices of symmetry reduction between two neighbouring subgroups are indicated by numbers in square brackets within these arrows. The index of a subgroup is the ratio of the number of symmetry elements in a group to that in the subgroup. These indices give the number of lower symmetry variants (domains) that can be generated when a transition from a high-symmetry to a low-symmetry structure occurs. Inclined arrows indicate symmetry changes due to atomic site (Wyckoff position) changes, leaving the occupancy fixed, i.e. displacive ordering. Vertical arrows, on the other hand, A

E

B A fA2

fC2

B

fS1 fA1

C

b

fC1

[010] fS2

D

F

C

[100]

a

Figure 5.21. The structure corresponding to Cmcm space group, with an appropriate choice of Wyckoff sites, can represent a structure which is close to hcp but differs from it in symmetry and in the relative positions of atoms on the basal plane.

426


Im3m (A2) [2]

[3] I4/mmm

Pm3m (B2) [3]

[2]

[2]

P4/mmm

Pmmm [2]

[2]

P 63 /mmc (A3) [2]

[2]

[4]

Cmcm (A20)

Cmmm [2]

[2]

P 63 /mmc (D019)

Pmmc (B19) [2]

[3]

Cmcm (A2BC)

Figure 5.22. The sequences of transitions connecting the higher symmetry cubic and hexagonal space groups to the lower symmetry orthorhombic space group. The directions of symmetry reduction are indicated by the arrows, and the indices of symmetry reduction between two neighbouring subgroups are indicated by numbers within these arrows.

indicate symmetry changes due to changes in atomic site occupancy, i.e. chemical ordering. Slight adjustments in site positions and occupancies due to new atomic environments will accompany chemical and displacive ordering, respectively. The sequences of possible structural transitions (Figure 5.22) are arrived at from the consideration of maximal subgroup relations. The atomic correspondence between these structures are depicted in Figure 5.23. The I4/mmm and Fmmm structures are obtained by homogeneous straining of the parent cubic lattice, A2 ¯ (Im3m). Similarly the P4/mmm and Cmmm structures correspond to a homoge¯ neously distorted B2 structure (Pm3m). While a tetragonal distortion along a cubic direction brings about the first step of the transition from the A2 and the B2 structures, the second step is associated with different distortions along two orthogonal cubic directions. The overall orthorhombic distortion of the cubic structure, if not accompanied by ordering, is probably unstable for materials with isotropic metallic bonding. Therefore these structures are not expected to exist as metastable states but rather represent a homogeneous strain accompanying the subsequent symmetry reduction by shuffle displacements (from Fmmm and Cmmm to Cmcm and Pmma, respectively). The Cmcm and Pmma space groups which correspond, respectively, to the Strukturbericht A20 (-U prototype) and B19 (AuCd prototype) structures can be obtained by heterogeneous shuffles of pairs of (110) planes of either a disordered


427

Im3m (A2)

011

a1

Cmcm (A20)

b

100

P 63 /mmc (A3)

a2

a

Ti, Al, Nb

Ti, Al, Nb

011

a1

Al, Nb

a2

Al Ti, Nb

Ti

100

a

Pm3m (B2)

b

Al, Nb

P 63/mmc (D019)

Ti

b

– Lower layer

Al Nb

– Upper layer Pmma (B19)

a

Ti

Cmcm (A2BC)

Figure 5.23. The sequences of transitions showing two branches, one starting from the disordered bcc structure and the other from the ordered B2 structure.

or an ordered cubic structure. The amplitude of the shuffle displacement wave is reflected in the y-coordinates of Wyckoff positions: 4c (0, y, 41 ) for the Cmcm structure and 2e ( 41 , y1 , 0); 2f ( 41 , y2 , 21 ) for the Pmma structure. For some special values of Wyckoff position’s y-coordinates and/or lattice parameters, a structure of higher symmetry can be generated. Such a high symmetry structure, hexagonal P63 /mmc, is generated from the disordered Cmcm structure 1 when the shuffles are such that √ the Wyckoff position parameter y is 3 and the ratio of lattice parameters, b/a, is 3. Such a symmetry enhancement is not possible for the orthorhombic Pmma structure due to the chemical order inherited from the B2 structure. On the other hand, no thermodynamic barrier exists for the Cmcm (A20) to P63 /mmc structural transition as the latter structure is associated with a higher entropy (due to its higher symmetry) while the internal energies, estimated on the basis of first near neighbour interaction energies, of the two structures are nearly the same. The structure with the lowest symmetry, the O-phase, has the Cmcm space group and ternary ordering on the Wyckoff positions 4c1 4c2 and 8g. This ternary ordering is responsible for making the translations in the O-phase structure twice as much as those in the binary ordered B19 structure. The O-phase structure can be obtained by ordering either the Pmma (B19) or the P63 /mmc (D019 )

428


structure. The D019 structure can be obtained by binary ordering of the disordered hcp (A3) structure and could correspond to a metastable intermediate phase. 5.3.5 Transformation sequences The group–subgroup relations suggest the following three transformation sequences, designated as A, B and C: ¯ A$ Im3mA2

12 CmcmA20 1 P63 /mmcA3 −−→ − → contd P63 /mmcA3 4 P63 /mmcD019 3 CmcmO − → − → ¯ ¯ B$ Im3mA2 2 Pm3mB2 12 PmmaB19 2 CmcmO − → −−→ − → ¯ C$ Im3mA2 12 CmcmA20 2 PmmaB19 2 CmcmO −−→ − → − → The numbers in the square brackets give the number of variants possible at each of the transition steps. It must be emphasized here that the A2 → O transformation can also occur in a single step provided the transformation mode is reconstructive. However, coherent transformations proceed in steps, following one of the sequences mentioned above. Each transient phase may exist over a temperature range between the upper and lower critical temperatures (or temperatures of phase instability for first order transformations). The number of variants in each transition is equal to the index of the subgroup (as indicated within the square brackets). For a sequence of transitions the number of variants of the lowest symmetry phase (with respect to the highest symmetry phase) will be equal to the product of the indices for each step. The creation of the expected number of variants at each transition state generates a hierarchy of domain structures. It is from the domain structure that the exact sequence of steps followed during the transformation can be identified. The microstructures resulting from the sequences, A, B and C, will all consist of the same O-phase but with different hierarchies and types of interfaces, and the nature of the interfaces, whether rotational, translational or mixed, will be determined by the group–subgroup relation. Each symmetry reduction step necessarily leads to the formation of more than one variant of the lower symmetry phase, thereby restoring the lost symmetry partially in the macroscopic sense. As mentioned earlier, the number of variants in each transition is equal to the index of the subgroup, indicated within the square brackets in the sequences. For the complete sequence of transitions the number of variants of the lowest symmetry O-phase is given by the product of the indices corresponding to each of the steps involved in the sequence. This means that the


429

sequence A results in 12 × 1 × 4 × 3= 144 variants while the sequences B and C result in 482 × 12 × 2 or 12 × 2 × 2 variants. A variety of domain interfaces can be created between adjacent domains. A domain interface will remain stress free if it is planar and if the self-strains on both sides are compatible, so that no discontinuity of displacements occurs at the interface. Bendersky et al. (1991) have examined the expected distribution of domains arising out of the different sequences, A, B and C, of the transformation and have listed the types of interfaces which can form between domains produced in different transitions (Table 5.13). The microstructural development corresponding to the three paths, A, B and C, has also been predicted on the basis of the domain configuration expected to be generated in the successive steps of the transformation. The three paths differ primarily as to whether the hexagonal symmetry phases or the B19 phase occur at an intermediate stage of transition. All the three paths finally lead to somewhat similar microstructures consisting of domains of the orthorhombic phase, although the microstructure at the intermediate steps are significantly different. For the transformation sequences A and C the first step, A2 → A20, is a displacive transition and involves the creation of a polytwin structure. In contrast, the first step of the sequence B, namely, the A2 → B2 transition, involves pure chemical ordering and produces only antiphase boundaries (APB). Polytwins are separated from their neighbours by planar boundaries, while the isotropic APBs, which are usually curved surfaces, separate either interconnected or closed volume domains. Table 5.13. List of interfaces between domains in different group/subgroup transitions. Group/subgroup

Type of interface

¯ ¯ Im3mA2 → Pm3mB2 ¯ → CmcmA20 Im3m

Translational APB Rotational twins Translational with stacking fault mixed twin/translational No new interface Translational APB Rotational (compound twins) Rotational twins Translational with stacking fault mixed twin/translational Translational APB Translational APB

CmcmA20 → P62 /mmcA3 P63 /mmcA3 → P63 /mmcD019 P63 /mmcD019 → CmcmO ¯ → PmmaB19 Pm3m CmcmA20 → PmmaB19 PmmaB19 → CmcmO

The interfaces are described by the domain generating symmetry operation of lowest symmetry.


A3

D019

O

B2

B19

O

A20

B19

O

→

430

→ →

A2

Figure 5.24. Schematics of development of microstructure by the creation of anisotropic planar interfaces (due to the associated stacking fault nature), isotropic APBs (resulting from pure chemical ordering) and secondary polytwin boundaries (as in the case of the D019 → O phase transition).

The B2 → B19 transition in the sequence B is responsible for the introduction of a polytwin structure. Further development of microstructure by the creation of anisotropic planar interfaces (due to the associated stacking fault nature), isotropic APBs (resulting from pure chemical ordering) and secondary polytwin boundaries (as in the case of the D019 → O phase transition) is schematically illustrated in Figure 5.24 The transformation paths predicted from group–subgroup relations pertain only to congruent phase transformations in which a single phase transforms into another single phase without involving a partitioning of the alloying elements. Such transformations in multicomponent systems can occur under equilibrium conditions only at special compositions, known as consolute points, or in second order transitions. However, under sufficient undercooling, a partitionless transformation can be induced. Bendersky and Boettinger (1994, Bendersky et al. 1994) have shown that the predictions made closely match the observed transformation sequences in some specific ternary Ti–Al–Nb alloys. The salient features of their experimental observations are summarized in the following, mainly in order to illustrate how observations on the domain distribution can lead to the identification of the transformation path. 5.3.5.1 Transformation sequence in the alloy Ti–25 at.% Al–12.5 at.% Nb The observations made on the needle-shaped transformation product in this alloy correspond to a situation where a displacive transition takes place in the disordered


431

A2 structure. Therefore, either of the sequences A or C can be expected to be operative. Bendersky and Boettinger (1994, Bendersky et al. 1994) have discussed the morphological features of the needle-shaped product formed in this alloy when cooled from 1373 K at the rate of 400 K/s and the displacement vectors associated with the domain boundaries contained within these needles. Even though diffraction patterns obtained from individual needles can be indexed in terms of one of the three variants of the D019 structure, the splitting of diffraction spots is indicative of the occurrence of orthorhombic distortions in the basal plane of the D019 phase, suggesting the presence of the O-phase. Two types of interfaces, namely, ¯ > and planar wavy, isotropic APBs with the displacement vectors, R = 16 < 1120 1 ¯ stacking fault-like defects with R = 4 0110, have been detected within these nee¯ > APBs points towards the occurrence of the dles. The presence of the 16 < 1120 disordered A3 phase as an intermediate step prior to the D019 ordering. Out of the ¯ > faults, only one variant has been actually three possible variants of the 41 < 1010 observed. This deviation from hexagonal symmetry suggests that the preferred direction (arising from orthorhombic symmetry) existed prior to the appearance of hexagonal symmetry, which is consistent with the initial step of sequence A. Finally the presence of O-phase domains, related to each other by hexagonal symmetry, is attributable to the occurrence of the last step of sequence A. 5.3.5.2 Transformation sequence in the alloys Ti–25 at.% Al–25 at.% Nb, Ti–28 at.% Al–22 at.% Nb and Ti–24 at.% Al–15 at.% Nb In both these alloys, Bendersky and Boettinger (1994) and Muraleedharan et al. (1992a,b) have provided evidences for the occurrence of A2 → B2 ordering prior to the transition to the close packed structure. This means sequence B is expected to be operative in these alloys which show self-accommodating plate-like domains. The observed lattice correspondence, [001]o [011]c and [100]o [100]c (o and c referring to the orthorhombic and the cubic phases, respectively), is consistent with the group–subgroup relations discussed earlier. This gives six rotational variants of the orthorhombic phase (either B19 or O), each with its basal (001)o plane parallel to one of the six {110}c planes of the cubic structure. Accommodation of transformation strains requires a small mutual rotation between the contacting variants by the creation of stress-free interfaces. The measured misorientation between adjacent plates has been shown to be consistent with that calculated from the lattice strains (Bain strain) required for the B2 → O transformation. An analysis of the displacement vectors associated with the domain boundaries has revealed the presence of two types of domain boundaries. The first type, with a wavy APB appearance, is associated with the displacement vector R = 21 [010]o (or the 21 [100]o vector which is equivalent due to the C-centring of the Cmcm space

432


group of the O-phase). APBs of this type are expected to form as a result of the B19 → O transformation in which the unit cell parameters, a and b, are doubled. The second type of domain boundaries show a distinct faceted appearance and correspond to the displacement vectors, R = 41 [012] or 41 [010]. These interfaces are expected to be produced during the B2 → B19 transition. The homogeneous (Bain) strain corresponding to the B2 → B19 transition is also responsible for the observed orientations of the six twin variants of the orthorhombic or the hexagonal structure. In view of these observations, it could be concluded that these alloys go through sequence B during relatively fast cooling from 1373 K. As mentioned earlier, the sequences predicted from the group–subgroup relations and those observed in the alloys discussed in this section correspond to the situation of congruent (partitionless) transformations. Transformation sequences pertaining to a situation where the cooling conditions allow alloy partitioning and phase reactions are discussed in the following section. 5.3.6 Phase reactions in Ti–Al–Nb system Phase reactions involving partitioning of the alloying elements can be understood using the proposed section (Banerjee et al. 1990) of the Ti–Al–Nb system (Figure 5.16) in the vicinity of the composition Ti2 AlNb. This pseudobinary section is along the composition line joining Ti3 Al and Nb3 Al. This section has later been modified and enlarged to some extent (Figure 5.25) by Bendersky and Boettinger (1994, Bendersky et al. 1994) by taking into account the following factors: (1) The disordered -phase definitely exists between the and the 2 -phases in pure Ti3 Al and the -phase field narrows down with increasing additions of Nb. β

Temperature (K)

1300

βo α

1200 1100

α2 + β

1000 900

α 2 + βo

α2

+β

+O

O

o

α2 α2 + O

800 700

10

Ti3Al

Ti–25Al–15Nb 20

Atomic % Nb

30

Nb3Al

Figure 5.25. The pseudobinary section, enlarged and modified, along the composition line joining Ti3 Al and Nb3 Al for the phase reactions involving partitioning of the alloying elements of the Ti–Al–Nb system in the vicinity of the composition Ti2 AlNb.


433

(2) The → o ordering transition qualifies to be a second order transition and therefore, the and o -phase fields need not be separated by a two-phase field. (3) The maximum stability of the o -phase is assumed to be centred around the Ti2 AlNb composition. This is because the two sublattices of the B2 structure of Ti–Al–Nb alloys are preferentially occupied by Ti atoms and by a mixture of Al and Nb atoms, respectively. The maximum order is, therefore, likely to be exhibited at a composition where the atomic fraction of Ti atoms is equal to the sum of the atomic fractions of Al and Nb atoms. The maximum in the ordering curve is taken to occur at about 1673 K (Bendersky and Boettinger 1989). (4) The → , → 2 and 2 → O transitions are all first order transitions; therefore, the , , 2 and O single phase fields are separated by two-phase fields. (5) The ternary phase diagram essentially depicts the phase stability regimes of the three phases, (2 ), (o ) and O as shown in Figure 5.25. The two-phase fields, 2 + o o + O and 2 + O, remain separated by a three-phase field, 2 + o + O. A variety of phase reactions is possible on cooling an alloy through the threephase field. These include the eutectoid reaction, o → O + 2 , the peritectoid reactions o + O → 2 and o + 2 → O, and the precipitation reactions, o → o + O, o → o + 2 and 2 → 2 + O. Out of these, the reactions that occur are determined by the alloy composition and the relative positions of the three phase isotherms at different temperatures. A systematic study of phase reactions in the alloy, Ti–25 at.% Al–15 at.% Nb, which passes through the three-phase field, has been reported by Muraleedharan et al. (1992a,b). The results of this study are summarized here (Figure 5.26) to illustrate the variety of phase reactions possible in such an alloy: (1) Quenching (cooling rate exceeding 100 K/s) from a temperature above 1383 K results in the retention of the o -phase, the /o transition temperature being around 1403 K. The → o ordering reaction cannot be suppressed by quenching. (2) Quenching from temperatures between 1253 and 1383 K produces a microstructure consisting of equiaxed 2 -grains in a o -matrix. Equilibration at a temperature below but close to 1253 K, followed by quenching, results in a three-phase microstructure comprising the 2 - and o -phases and a blocky O-phase. The transus temperatures bounding different phase fields for this alloy have been identified from these observations and are indicated by the

434


Ti–25Al–16Nb

T-T-T curves, continuous cooling β B2 α2 + B2

B2 → α2 + B2

Temperature (°C) →

1000 α2 + α′2 + O

α2 + B2→O α2 + B2 + O

800

O + B2 B2 → O + B2

10°C/s 4°C/s AC AQ

0.7°C/s FC I

0.1°C/s FC II

0.02°C/s FC III

600

10

102

103

104

Time (s) →

Figure 5.26. Phase reactions in the three-phase field in Ti–24 at.% Al–15 at.% Nb ternary alloy.

intersections of the dotted vertical line corresponding to the alloy composition with the o /2 + o and 2 + o /2 + o + O transus lines. (3) Continuous cooling at slower rates (10–0.02 K/s) results in the formation of a primary phase – either O or 2 – depending on the cooling rate. Phase reactions induced under different cooling rates are indicated in the schematic T-T-T diagram shown in Figure 5.27. The kinetics of the o → 2 + o and the o → O + o reactions are such that faster cooling rates result in the -phase transforming directly to the O-phase, skipping the formation of the 2 -phase, even though the 2 + o and 2 + o + O phase fields are crossed during the continuous cooling process. (4) In case the 2 -phase is present as a primary decomposition product, a variety of secondary reactions such as 2 → 2 +O and 2 +o → O are possible. The fine multivariant O-phase plates observed at the periphery of the 2 -plates, which satisfy the 2 /O orientation relation can be identified as the product of the former phase reaction. On the other hand, the monolithic O-phase rim produced around the 2 -plates corresponds to the product of the peritectoid reaction between the 2 - and o -phases. The various microstructures that can be produced by continuous cooling in the Ti–24 at.% Al–15 at.% Nb alloy under different rates of cooling are shown in Figure 5.27 to illustrate the variety of phase reactions observed.


435

Figure 5.27. Various microstructures produced by continuous cooling in the Ti–24 at.% Al–15 at.% Nb alloy under different rates of cooling (after Banerjee, 1994a).

Ageing of the -quenched alloys in the O + o and the 2 + o phase fields produces 2 - or O-phase precipitates in the form of Widmanstatten laths in a o matrix, thin films of the retained o -phase being left behind at the lath interfaces. Samples aged after subtransus solution treatments show changes with regard to the primary 2 -as well as the metastable o -phases. Ageing in the O + o phase field results in the decomposition of the o -phase through two different kinetically competitive modes which operate simultaneously; the first is a conventional continuous precipitation of the O-phase in the o -phase while the second results in o -precipitation within O-phase grains. The mechanism of the o → O transformation has been discussed in detail in a later section. Based on the phase diagram (Figure 5.25) discussed in this section, a To versus composition diagram (Figure 5.28) can be constructed, To being the temperature where the parent and the product phases have the same molar free energy. Such diagrams are useful in rationalizing composition-invariant transformations, which have been described in Section 5.3.5. A schematic drawing (Figure 5.29) showing the free energies of the /o , /2 /O phases as functions of composition has been constructed (Bendersky and Boettinger 1994, Bendersky et al. 1994). This diagram can be viewed as the superposition of the free energy–composition plots for ordering transformations in the bcc-based and hcp-based phases.

436

Phase Transformations: Titanium and Zirconium Alloys β (bcc)

1300

α (hcp)

1100

α2 (D019)

B2

900

α

1000

β→

β→

To (°C)

1200

βo (B2)

O α2 → O →

θ → B2 9 B1

B2

800 700

Ti3Al

10

20

30

Atomic % Nb

Nb3Al

Figure 5.28. To versus composition diagram (based on the phase diagram in Figure 5.25) to rationalize composition-invariant transformations.

Free energy

α

2

Alloy 2

α

Alloy 1

β βo B19

O

Ti3Al

10

20

Atomic % Nb

30

Nb3Al

Figure 5.29. A schematic drawing showing the free energies of the bcc-based /o and hcp-based /2 /O phases as functions of composition.

5.4

FORMATION OF ZR3 AL

The equilibrium Zr3 Al phase has the L12 (cP4) structure. The availability of several slip systems in the L12 structure, which is essential for meeting the Von Mises criterion for ductility of a polycrystalline material, made it attractive to consider Zr3 Al as an intermetallic for structural applications. The equilibrium Zr3 Al phase differs from the equilibrium Ti3 Al phase in the following subjects: (a) Unlike Ti3 Al (D019 structure), the Zr3 Al does not form by chemical ordering of the terminal


437

solid solution (hcp -phase). (b) The Zr3 Al phase forms through a peritectoid reaction, + Zr 2 Al → Zr3 Al (refer phase diagram given in Chapter 1). (c) The Zr3 Al phase is a line compound which does not tolerate any significant deviation from the stoichiometric composition. 5.4.1 Metastable Zr3 Al (D019 ) phase Mukhopadhyay et al. (1979) explored the possibility of forming a metastable Zr3 Al phase with the D019 structure by quenching an alloy of composition Zr–14 at.% (4.6 wt%) Al from the -phase field. Water-quenched thin (0.5 mm thickness) samples of thin alloy showed a lath martensite structure. Selected area diffraction experiments have shown the presence of diffuse intensity maxima at D019 superlattice positions apart from the hcp reflections. On subsequent ageing, these superlattice reflections have been found to intensify and sharpen. Dark-field images with these superlattice reflections have shown the presence of 5–10 nm microdomains of the D019 phase distributed in the martensite matrix (Figure 5.30). This observation has clearly established that in a supersaturated Zr–Al solid solution (with hcp structure), a metastable D019 phase can form. This is quite expected as the development of the D019 structure in an hcp lattice can be achieved by replacive ordering. Figure 5.31 shows how by an introduction of a concentration wave with the wave vector, k= a2 1120 , the D019 structure can be formed in an hcp lattice. As has been mentioned earlier, the wave vector satisfies two of the three

Figure 5.30. (a) Dark-field image showing the presence of 5–10 nm microdomains of the D019 phase distributed in the martensite matrix and (b) corresponding D019 diffraction pattern along the (0001) direction.

438

Phase Transformations: Titanium and Zirconium Alloys r

ye

La

0

L 1

0.5

2

2d1120 t1 < t2

4

A layer B layer

Al Zr Al Zr

Al fraction

3

t1

0.25

t2

p=0

1

2

3

4

¯ planes. Figure 5.31. (a) Atomic arrangement of the D019 structure showing compositions of {1120} (b) Development of concentration wave as a function of time (t) leading to the formation of D019 structure from the parent hcp phase (average composition Ti–25 at.% Al).

Landau–Lifshitz conditions for the A3 → D019 transformation to be a second order transition. This is indeed a fit case for a continuous ordering under a high degree of supercooling. Figure 5.31 illustrates the evolution of the A3 → D019 ordering by formation and amplification of a concentration wave with k= a2 1120 in an alloy of stoichiometric composition, i.e. Zr–25 at.% Al. Supersaturation of the -phase to such an extent is difficult, if not impossible. In the Zr–14 at.% Al alloy, investigated by Mukhopadhyay and Banerjee (1979), the attainment of near-stoichiometric composition has been achieved by a spinodal clustering process which has resulted in a concentration modulation of wavelength of about 20 nm. It is reported that the first step of the transformation has been the spinodal clustering which resulted in a concentration-modulated structure along 1010 direction, the elastically soft direction in the -phase. The regions getting enriched with Al subsequently undergo an ordering process by developing and amplifying a short wave length concentration wave. The overall process can, therefore, be considered as a superposition of spinodal clustering followed by continuous ordering. Such a transformation sequence can occur if the system at the ageing temperature is initially unstable with respect to a long wave length (20 nm) concentration fluctuation and, after attainment of a concentration within the ordering spinodes, becomes unstable with respect to the A3 → D019 ordering (within the Al-enriched regions). The condition under which such a concomitant ordering and clustering processes can occur is shown in the schematic free energy–concentration plots (Figure 5.32) corresponding to the disordered () and the ordered (, D019 ) phases. An alloy of composition C1 (which lies between the spinodes of free energy–concentration plot for the disordered -phase) is unstable with respect to spinodal clustering (concentration modulation with wave vector, k = 000 ). This process leads to the development of alternate Al-rich regions which in turn becomes unstable with respect to → D019 ordering. With the gradual amplification of 21 1120 concentration wave, the order


439

T3

Free energy →

D

A

A

A x

x

Composition (% B) →

Figure 5.32. A schematic free energy–concentration plot depicting the condition under which concomitant ordering and clustering processes can occur corresponding to the disordered, , and the ordered, D019 , phases.

parameter in these regions increases finally leading to nucleation of Zr3 Al particles of D019 structure. These ordered particles decorate the alternate Al-enriched layers, as shown in Figure 5.30. 5.4.2 Formation of the equilibrium Zr3 Al (L12 ) phase The equilibrium Zr3 Al (L12 ) evolves from the -quenched structure containing a distribution of fine 2 (Zr3 Al with D019 structure) particles in the -matrix. The emergence of the equilibrium Zr3 Al phase has been found to occur through a cellular precipitation process. The nucleation of the two-phase cells ( + Zr3 Al) invariably occurs at the lath boundaries of the -quenched structure. These cells grow by propagation of the cell boundaries towards the interior of laths (Figure 5.33). Some of the general characteristics of cellular precipitation can be illustrated by this example. These are listed as follows: (1) Nucleation of a cell (two-phase region containing a group of and Zr3 Al (L12 ) lamellae, each with a specific orientation) invariably occurs at the grain boundaries. In the present example, the starting structure is that of a lath martensitic -matrix with a distribution of fine Zr3 Al (D019 ) precipitates. Majority of the lath boundaries are small-angle boundaries separating laths of close orientations.

440


(a)

(b)

(c)

Figure 5.33. Micrographs showing a cellular precipitation process. (a) The nucleation of the twophase cells ( + Zr 3 Al) invariably occurs at the lath boundaries of the -quenched structure. (b) Growth of the cells by propagation of the cell boundaries towards the interior of laths. (c) Twophase ( + Zr 3 Al) transformation product.

(2) The -orientation (1 ) of the cell matches with one of the neighbouring -lath (1 ), while the Zr3 Al (L12 ) orientation maintains exact orientation relationship with 1 . (3) A cell so related grows into the adjacent lath (orientation 2 ). Similarly a cell with 2 -orientation grows into the 1 -lath. Such a process essentially makes the original flat lath boundary into an undulated boundary which acts as the transformation front, the propagation of which accomplishes the transformation of the + Zr 3 Al (D019 ) region into + Zr 3 Al (L12 ) cells (Figure 5.34). (4) In general, where a supersaturated phase decomposes into a cellular producer, the transformation can be described as a phase reaction, → + , where is the precipitating phase. (5) The process of partitioning of the alloying elements between the two product phases occurs primarily of the transformation front. The diffusion field is, therefore, related to the lamellar spacing of the product structure.

Figure 5.34. Micrograph showing periodic spacing of interfacial dislocations at /Zr3 Al boundaries and faults within Zr3 Al lamellae.


441

Interfacial Dislocations

α

(0002)α // (111) Zr3Al

α

Zr3Al

m c

Figure 5.35. A schematic drawing showing the misfit dislocations along the interface between and Zr3 Al (L12 ) phases.

(6) The diffusion distance being somewhat small and the diffusion being enhanced due to the presence of the interface (transformation front), the kinetics of cellular precipitations are relatively fast. (7) In the present case, where the + Zr 3 Al (D019 ) structure decomposes into the equilibrium +Zr 3 Al (L12 ) structure, it is imperative that the metastable Zr3 Al (D019 ) precipitates get dissolved ahead of the transformation front leading to an increase in the level of supersaturation in the -phase which breaks up into the equilibrium + Zr 3 Al (L12 ) phase mixture at the transformation front. As mentioned earlier, the orientation relationship between the and the Zr3 Al (L12 ) phases is maintained within the cells. The lamellar product, therefore, contains primarily two types of semicoherent interfaces characterized by arrays of misfit dislocations. The degree of misfit along the c-direction of the phase being small, the spacing between misfit dislocations is quite large. A schematic drawing (Figure 5.35) shows the misfit dislocations along the interface between and Zr3 Al (L12 ) phases. 5.4.3 +Zr2 Al → Zr3 Al peritectoid reaction The phase diagram (Figure 6.30) shows that the equilibrium Zr3 Al (L12 phase) results from a peritectoid reaction, + Zr 2 Al → Zr3 Al. The Zr2 Al (B82 structure) can be viewed as an ordered #-phase, as described in detail in Chapter 6. Due to the close lattice matching between and Zr2 Al phases, the latter grows in the -matrix as equiaxed particles as an alloy is cooled in the + Zr 2 Al phase field.

442


← Tp

Temperature (K)

1200

← Tα 1100

1000

80%

900

50% 5% 800

0.1

1

10

100

1000

Time (h)

Figure 5.36. The proposed time-temperature-transformation (T-T-T) diagram showing that the rate of the peritectoid reaction is fastest at around 888 C, where a Zr2 Al particle of about 3 !m diameter gets fully transformed within about 8 h.

The average size of Zr2 Al particles depends on the rate of cooling through the two-phase field and varies from 8–10 !m in diameter in larger ingots to about 1 !m in the smaller ones. When the + Zr 2 Al phase mixture is allowed to react at a temperature below about 975 C, the two phases undergoes a peritectoid reaction resulting in the formation of the Zr3 Al (L12 ) phase. Schulson (1975) and Schulson and Graham (1976) have studied the peritectoid reaction in detail. The time-temperature-transformation proposed by them (Figure 5.36) shows that the rate of the peritectoid reaction is fastest at around 888 C, where a Zr2 Al particle of about 3 !m diameter gets fully transformed within about 8 h. Since peritectoid reaction requires the presence of two reacting phases, the reaction invariably initiates at the interface. The reaction product, in this case, Zr3 Al “envelopes” the Zr2 Al particles and provides a barrier for the reaction to proceed rapidly. The subsequent growth of the product layer requires diffusion of the fast moving species (in this case Al) across the product layer. Schulson and his coworkers have clearly shown the presence of the envelopes of Zr3 Al phase around Zr2 Al particles and have shown that growth of the product phase occurs through long-range diffusion-controlled migration of Zr2 Al/Zr3 Al and Zr3 Al/-Zr interfaces in opposite directions. Correspondingly, the transformation time is strongly dependent on the size of the Zr2 Al particles in the starting microstructure.


5.5

443

PHASE TRANSFORMATION IN -TiAl-BASED SYSTEMS

Two-phase -TiAl-based alloys offer an attractive combination of properties, namely, low density, as well as good creep resistance, high-temperature strength and oxidation resistance. It is because of the excellent combination of several desirable properties that these alloys are being considered as candidate materials for high-temperature applications in aerospace industries, particularly in gas turbine engines. Monolithic -TiAl is very brittle at room temperature. Several attempts have been made to improve the room temperature ductility of TiAl-based alloys by controlling the alloy composition and by tailoring the microstructure through phase transformations. The evolution of microstructure in these alloys is somewhat unique and is strongly influenced by the orientation relation between the participating phases. The two-phase lamellar structure, which is exhibited by these alloys when appropriately heat treated, is responsible for imparting to them an attractive combination of mechanical properties. The optimization of the mechanical properties requires a proper control of the sizes and volume fractions of the lamellar colonies and the interlamellar spacing. Another important aspect of these alloys is their microstructural stability, which determines the service life of components made from these during exposure to elevated temperatures. Phase transformation studies in these alloys, therefore, address two main issues: (a) the evolution of microstructure through different sequences and mechanisms of transformation and (b) the stability at elevated temperatures of the microstructure so obtained. 5.5.1 Structural relationship between 2 - and -phases In binary Ti–Al alloys in the composition range of 38–48 at.% Al, two ordered intermetallic phases, 2 -(Ti3 Al) and -(TiAl), coexist in equilibrium at temperatures below about 1270 K. The D019 structure of the 2 -phase has been described earlier. The -TiAl phase possesses a tetragonal L10 structure. The lattice parameters of the 2 -phase (with 38 at.% Al) and the -phase (with 48 at.% Al) are: 2 $ a = 057 nm% c = 046 nm% c/a = 0803 $ a = 040 nm% c = 040 nm% c/a = 1020 The axial ratio of the tetragonal unit cell of the -phase being very close to unity, the structure of the -phase can be approximated to be cubic and the transformation between the 2 - and the -structures can be considered as that between an ordered hcp and an ordered cubic structure. Figure 5.37 also shows the atomic arrangement and the interatomic distances on the basal (0001) plane of the D019 structure and on the close packed (111) plane of the L10 structure. It may

444

Phase Transformations: Titanium and Zirconium Alloys (0001) of α2-phase

(111) of γ-phase

Ti atom

Al atom

Figure 5.37. Atomic arrangement and the interatomic distances on the basal (0001) plane of the D019 structure and on the close packed (111) plane of the L10 structure.

be noted that the interplanar spacings of the (0002)2 and the (111) planes are very close, the difference being smaller than 0.5%. Again, the nearest neighbour distances in the close packed planes in these two phases differ only by about 2%. The near-perfect lattice matching between the close packed planes of the 2 (D019 ) and (L10 ) phases is the determining factor which controls the morphology of the lamellar morphology of the 2 + structure. A strict orientation relation between 2 and , which is invariably maintained in the lamellar 2 + structure, is revealed from the diffraction pattern shown in Figure 5.38. This orientation ¯ > < 110 > , first reported by Blackburn relationship, (0001)2 {111} ; < 1120 2 (1967), conforms to the commonly observed orientation relationship in fcc to hcp (002)γ

T

(2200)α

(002)γ

2

(220)γ

(2202)α

T

2

(111)γ (0002)α

(111)γ

000 2

(111)γ

T

γ fundamental γT fundamental α2 fundamental

(111)γ

T

(220)γ

γ superlattice γT superlattice α2 superlattice

Figure 5.38. Diffraction pattern showing orientation relation between 2 and , in the lamellar ¯ 110

¯ . 2 + structure. Zone axis: 1120

2


445

structural transitions. The diffraction pattern (Figure 5.38) shows reciprocal lattice sections of two twin-related variants, the twinning plane being the specific {111} plane which is parallel to the basal plane of 2 . The lamellar aggregate of the - and 2 -phases can also be viewed as a stacking of close packed layers in the cubic ABCABC sequence over a certain distance corresponding to the thickness of the -plate, followed by a hexagonal stacking sequence of ABAB which builds up the 2 -structure. The diffraction pattern shown in Figure 5.38 suggests that a group of neighbouring lamellae contains regions not only corresponding to ABCABC stacking but also to ACBACB stacking, the two -regions being twin related across the {111} plane. Twin-related -plates can either remain in contact along the twin plane or may be separated from each other by 2 -plates. Single -plates are also seen to be divided into ¯ and [011] ¯ ordered domains which are generated due to the fact that the [110] directions of the L10 structure are not crystallographically equivalent. The possible orientation variants of 2 + structure are shown in Figure 5.39.

110

101

011

101

101

OR1

[011]

110

011

[101]

1120

110

011

011 OR2

OR6 1210

2110

011

α2

[110]

110

101

110

OR3

OR5

(a)

110

101

011

101 Ti Al

OR4 = γI (ABC-type stacking) = γII (ACB-type stacking)

(b)

Figure 5.39. The possible orientation variants of 2 + structure.

446


5.5.2 Phase reactions The relevant portion of the binary Ti–Al phase diagram, which is based on the work of Perepezko and Mishruda (1993), is shown in Figure 5.40. It can be seen from the phase diagram and the transformation pathways indicated by vertical dashed lines that the 2 - and the -phases can evolve through one of the pathways listed below: (1) (2) (3) (4) (5)

→ → 2 → 2 → 2 + → + → 2 + → → 2 +

The phase evolution sequences reported in Ti–Al alloys containing about 35– 50 at.% Al (Jones and Kauffman 1993, Yamabe et al. 1995) are discussed in the following. 5.5.2.1 Ti-34-37 at.% Al; → → 2 As these alloys are cooled from the -phase field to the 2 -phase field, one of the three transformation processes takes place depending on the cooling rate.

1600 L α+L

β+L

1500

Temperature (°C)

β 4

α+β

1400 1

2 α

1 3

1300

1100

α+γ

α + α2

1200 α2

γ

5

α2 + γ

1000

35

40

45

50

Atomic % Al

Figure 5.40. A portion of the binary Ti–Al phase diagram depicting the transformation pathways, as indicated by vertical lines, for evolution of 2 - and the -phases.


447

These are: precipitation of either - or 2 -plates in the -matrix involving longrange diffusion, composition-invariant massive transformation and diffusionless martensitic transformation. Precipitation of the -phase in the -phase matrix produces a basket weave microstructure akin to that commonly encountered in ( + ) Ti alloys. In contrast, a faster cooling induces a composition-invariant massive transformation in which massive -grains with jagged boundaries are produced. Al being a strong stabilizer, the To for the → transformation is about 1523 K. At such a high temperature the kinetics of massive transformation are expected to be very rapid. It is, therefore, difficult to suppress this massive transformation by any solid state quenching technique. The → 2 ordering temperature is also very high (between 1373 and 1473 K) and, therefore, the ordering process immediately follows the formation of the -phase, which forms either as Widmanstatten -plates or as massive -grains. 5.5.2.2 Ti-38-40 at.% Al; → 2 → 2 + The alloys which follow the transformation sequences (2) and (21 ) generally belong to hypo- and hyper-eutectoid compositions, respectively. Before discussing these transformation sequences it is worthwhile to consider a few special features of the eutectoid reaction → 2 + . As has been discussed earlier, the 2 -phase (D019 structure) is an ordered version of the -phase (hcp, A3 structure). The compositional proximity of the -phase of eutectoid composition (39.5 at.% Al) and the equilibrium 2 -phase (38 at.% Al) suggests that the extent of supersaturation required for a congruent ordering to occur is only marginal. This ordering transformation, occurring at a temperature in the range of 1373–1473 K, proceeds very fast without the requirement of any significant undercooling. The formation of the -phase from the -phase requires a change in the stacking sequence though the lattice matching of the two structures on (0001)2 (111) planes is excellent. In the case of a composition-invariant → transformation the structural change can be brought about by the passage of Shockley partials on every alternate close packed plane. This mechanism will be discussed in the next section. Such a lattice correspondence makes the kinetics of the lengthening of -plates quite fast but the kinetics of their thickening is rather slow. The overall kinetics of -phase formation is, therefore, much slower than that of the → 2 transition. In view of this, it is quite unlikely for the high-temperature -phase to decompose by a pearlitic mode in which the two product phases simultaneously emerge from the parent phase across the incoherent transformation front. The non-pearlitic modes of eutectoid decomposition which can operate in hypo- and hyper-eutectoid Ti–Al alloys are outlined in the transformation pathways (2) and (21 ), respectively.

448


An alloy in the composition range Ti-38-40 at.% Al, when cooled from the -phase field to a temperature below the eutectoid, first crosses the To temperature for the → 2 ordering. As explained earlier, the equilibrium → 2 + eutectoid decomposition is kinetically quite unfavourable in comparison with the pathway → 2 → 2 + . The decomposition of the -phase in a pearlitic mode has, therefore, not been encountered in experiments. The fact that → 2 ordering precedes the formation of -plates is revealed from the observation that -plates cut across APBs in the 2 -phase. A typical time-temperature-transformation diagram for alloys containing 38–40 at.% Al has been schematically illustrated in Figure 5.41(a) which depicts the kinetics of the competing processes on a relative scale. Since the precipitation of the -phase from the 2 -phase involves an incubation period of nearly 100 s, it is possible to suppress the transformation by a rapid quench which produces a metastable 2 -structure. On subsequent ageing -precipitation occurs within the 2 -grains, producing a lamellar 2 + structure.

5.5.2.3 Ti-41-48 at.% Al; → + → 2 + In these hypereutectoid alloys, -phase precipitation in the -phase is the primary step that occurs during slow cooling from the -phase field. Again the same lamellar + morphology, dictated by the Blackburn orientation relation, is exhibited. Subsequently the -regions undergo the ordering process during cooling through the To temperature for → 2 ordering. At a somewhat faster cooling rate, the step comprising -phase precipitation in the -phase is skipped; the → 2 ordering occurs first which is followed by the decomposition of 2 into the 2 + lamellar mixture. During a still faster cooling, the formation of the -phase can be completely suppressed, resulting in the retention of a metastable 2 -product in the as-quenched condition (as shown in Figure 5.41(b). However, suppression of -phase formation by quenching is not possible in alloy compositions close to the -phase field. A competing → massive transformation comes into play in such alloys. For example, an alloy containing 48 at.% Al, in the as-quenched condition, shows only a small volume fraction of lamellar 2 + regions while the major constituent of the microstructure comprises large -grains which contain stacking faults, twins and sometimes APBs. The nature of these defects has been analysed in detail mainly for ascertaining whether the → transformation goes through an intermediate disordered fcc phase. This point will be taken up later while discussing the mechanism of the massive → transformation, which was first reported by Wang and Vasudevan (1992) and later confirmed by Denquin and Naka (1995) and Zhang and Loretto (1995). Figure 5.41(c) shows a time-temperature-transformation

Temperature (°C)


449

Eutectoid α → α2

α → α2 + γ

α2 → α2 + γ

α2 + γ (fine γ plates)

α2 (thermal APBs)

Log time (a) α /α + γ

Temperature (°C)

1200

α2 → α + γ

1090

Eutectoid α /α + α2

α2 → α2

Metastable extension

α2 → α2 + γ α2 (thermal APBs) α2 + γ (lamellar)

Log time (b) 1500

Temperature (°C)

b

Tα

1400

γLamellar

TO

1300

γFeathery

1200

γMassive

Tγ

1100 1000

100°C/s 200°C/s 300°C/s

500°C/s

900

1000°C/s

400°C/s

800

0.1

1

10

Time (s)

(c)

Figure 5.41. (a) A schematic time-temperature-transformation (T-T-T) diagram for alloys containing 38–40 at.% Al depicting the kinetics of the competing processes on a relative scale. (b) T-T-T diagram showing kinetics of (i) → + and (ii) → 2 → 2 + processes and (c) T-T-T diagram indicating a kinetic competition between the processes, → + (lamellar or feathery) and → (massive).

450


diagram indicating a kinetic competition between the processes → + and → (massive). 5.5.2.4 Ti-49-50 at.% Al; → Alloys in this narrow composition range can transform into a single phase structure under the equilibrium condition. Two distinct types of product microstructure have been encountered; one is described as Widmanstatten colony and the other as massive (Yamabe et al. 1995). In the former case, the original -grains are divided into or -like regions with different orientations. Subsequently these regions transform into a lamellar microstructure containing primarily -plates. The mechanism of and the motivation for the creation of -regions of different orientations have not been identified so far. The second mechanism of the → transformation, namely, massive transformation, operates at faster cooling rates under which Widmanstatten colony formation is suppressed. 5.5.2.5 Ti49-50 at.% Al; → When alloys in this composition range, after solutionizing in the single phase field, are brought to the + phase field, -plates precipitate. The cubic symmetry of the parent -phase allows the formation of different variants of -plates along the {111} habit. Some illustrative examples of different transformation products in -TiAl-based alloys are shown in Figure 5.42. The formation of the lamellar microstructure and of the massive -phase are the two most important phase transformation phenomena encountered in the -TiAl alloys. In addition, discontinuous coarsening of the lamellar structure plays a major role in the evolution of microstructure. The suggested mechanisms underlying these phenomena are summarized in the following section.

100 μm

(a)

10 μm

(b)

Figure 5.42. Illustrative examples: (a) massive m products in Ti–49 at.% Al alloy and (b) two-phase lamellar 2 + structure observed in light microscopy.


451

5.5.3 Transformation mechanisms 5.5.3.1 Formation of the 2 + lamellar microstructure The overall transformation process for the → 2 + decomposition can be viewed as consisting of the ordering reaction → 2 and the precipitation reaction → . Ordering precedes or follows precipitation depending on whether the decomposition temperature is below or above the To temperature for ordering. In both the cases, the ordered 2 -superlattice is created on the hcp lattice of the -phase by the required atomic replacements. The lattice correspondence between the A3 and D019 structures, in which their respective basal planes remain parallel, allows the formation of only a single rotational domain. However, APBs corresponding to translational domains associated with displacement vectors of the ¯ > are formed. Such APBs within the 2 -phase are usually known type 13 < 1120 as thermal APBs. The precipitation of the -phase from the -phase involves both structural and chemical changes. The former is accomplished mainly by a transition in the stacking sequence from the hexagonal to the cubic type and the latter requires preferential Al enrichment of the regions where -precipitates appear. These precipitates invariably maintain lattice registry with the parent crystal and the precipitate plates have a strong tendency to form into a group. The kinetics of the lengthening of the -plates along the (0001)2 habit is much faster than that associated with their thickening. Groups of -plates often appear in the vicinity of grain boundaries and occasionally within the /2 grains. The habit plane being the basal plane, only one habit plane of -variant can form within a given 2 -grain. With the growth of these -precipitates, every single grain of the parent /2 structure gets transformed into a lamellar 2 + structure with a fairly uniform interlamellar spacing. The nucleation of -precipitates in a group and the uniform spacing between the -lamellae suggest that the formation of -plates and their growth do not take place individually but in a cooperative manner, the concentration field ahead of the edge of the plates being responsible for deciding the lamellar spacing. The most important difference between this mechanism and that of a cellular growth (as in the case of discontinuous precipitation or pearlitic transformation) lies in the fact that the lamellar product is not separated from the parent phase by an interface (Figure 5.43). This is because of the close lattice matching between the advancing -lamellae and the parent /2 matrix across the transformation front. This point is illustrated schematically in Figure 5.43. Transmission electron microscopy (TEM) studies have revealed some important features of the nucleation and growth steps of -phase precipitation. Blackburn (1970) and Sundararaman and Mukhopadhyay (1979) have shown that the nucleation of the -phase occurs heterogeneously, solely by the dissociation of a ¯ > dislocations on the basal plane. The stacking faults enclosed by the < 1120 6 2 a ¯ > partials can be considered as single layer nuclei of the -phase. < 1010 6 2

452


→

→

→

(a)

(b) Transformation front Alloy partitioning α /α2 Replacement ordering α /γ transformation chemical ordering + change in stacking sequence

γ

α2

(c)

Incoherent boundary No lattice site correspondence Cellular transformation

(d)

→

(e)

(f)

Figure 5.43. Schematics of difference between mechanism of lamellar and of a cellular growth as in the case of discontinuous precipitation or pearlitic transformation.

Preferred sites for such nucleation are grain boundaries, subgrain boundaries and individual dislocations. The density of these faults has been found to increase as the cooling rate from temperature above the 2 / + 2 transus is decreased. The growth of these nuclei to -plates can be compared with the hcp → fcc transformation in so far as the stacking sequence is concerned. The passage of a ¯ > partial dislocations over every alternate basal plane produces the < 1010 6 2 cubic stacking sequence. One of the possible modes by which a single layer stacking fault can grow into a -plate of several tens of nanometre thickness is by the operation of a pole mechanism in which a Shockley partial dislocation is anchored to a sessile [0001]2 pole dislocation. The rotation of the Shockley partial will change the stacking sequence, and during rotation the partial climbs along the pole through a distance equal to twice the interplanar spacing; thus each


453

rotation causes the product phase to thicken by two atom layers. Mukhopadhyay (1976) has provided some evidences for the presence of spiralling Shockley partial dislocations on the basal plane of the 2 -phase. Partitioning of Al atoms between the 2 - and -phases must occur through diffusive jumps in order to allow the -plates to attain the equilibrium composition. The movement of Shockley partials and the Al enrichment of the faults possibly take place concurrently, as the latter is expected to reduce the stacking fault energy. No doubt it is difficult for individual atoms to join the -lamellae on the broad face which is a highly stable low-energy planar interface between the 2 and -lamellae. In contrast, lateral growth of -lamellae through the movement of Shockley partials on the habit plane by the so called “terrace-ledge-kink” mechanism is a plausible mode in which the appropriate composition change of a growing lamella is ensured by the transfer of atoms on to kinks which provide favourable sites for atomic attachment. The fact that a combination of shear and diffusional processes is operative for the growth of -lamellae has been clearly revealed by high-resolution electron microscopy studies (Huh et al. 1990) on the 2 / interfaces which show a lateral ¯ > dislocations and by atom probe investigagrowth by the passage of a6 < 1010 2 tions of Denquin and Naka (1995) showing the sluggish nature of the diffusional growth. The lamellar -precipitates formed during fairly rapid cooling exhibit non-equilibrium partitioning of Al in the 2 - and -phases, the former remaining supersaturated in Al and the volume fraction of the latter being much larger than that corresponding to the equilibrium condition. The compositions of these two phases and their volume fractions approach the equilibrium values on ageing. It is to be noted that the results of the atom probe experiments indicate the absence of any concentration gradient around the 2 / interfaces; this suggests that the lamellar growth is controlled by the ledge mechanism rather than by a classical long-range diffusional process. The growth rate slows down with decreasing Al supersaturation of the matrix which reduces the driving force for the ledge-kink motion on the broad face. 5.5.3.2 Mechanism of the → massive transformation Ti–Al alloys (48–50 at.% Al) which are quite close to the -phase field, when fast cooled from the -phase field, transform into the single -phase. This transformation takes place without any change in composition and the morphology of the product phase has several features which suggest that the transformation is of the massive type. Wang and Vasudevan (1992) were the first to report the occurrence of such a massive transformation in this system and this observation has been confirmed in a number of more recent studies (Jones and Kauffman 1993, Zhang and Loretto 1995, Denquin and Naka 1995, Wang et al. 1998). The

454


-phase produced by this massive transformation is generally designated as m in order to distinguish this phase from the lamellar -phase. The quenched structure of alloys containing 48–50 at.% Al often shows a mixed morphology with massive m -regions coexisting with lamellar 2 + regions. The orientation of the 2 -lamellae in a given region gives the orientation of the prior -grain, as the -and 2 -structures are related by a unique orientation relation. Attempts aimed at finding out a possible orientation relation between the prior -grain and the m -product have not been successful (Denquin and Naka 1995), implying that no specific orientation relation exists between them. The interfaces separating adjacent grains of the m -phase and those between m - and matrix 2 phases have been found to be quite jagged and it is clear that such a morphology cannot result from a shear-like process in which a planar contact between the parent and product phases is expected. The massive m -grains consist of a large number of faulted domains. The defects present in these domains are of various types: stacking faults lying on all the four variants of {111} planes, microtwins and APBs which do not adhere to any crystallographic plane. The displacement vectors associated with these APBs correspond to a2 vectors. A detailed contrast analysis of the defects in the m -phase (Wang et al. 1998) has revealed that both 21 x > x1 , on quenching from the -phase field, exhibits the formation of the athermal -phase which inherits the composition of the parent -phase. This metastable product undergoes compositional changes during subsequent ageing when -particles depleted in the -stabilizing element form, replacing the athermal -particles. The composition, x2 , corresponds to the limiting composition for the formation of the athermal -phase. Beyond this composition, the -phase is retained on quenching. The composition of the aged -phase is given by the metastable boundary between the - and the + phase fields. Schematic free energy versus composition plots corresponding to the -, the - and the -phases (Figure 6.1(b)) illustrate the relative stabilities of these phases. Construction of

476


x

x

Equilibrium α /β Metastable ω /β Ms (α) Ms (ω) Athermal ω Thermal ω

x

x

x

↑

T

x

a

c

e f

d b

x

T1

x

298 K

↑ Gα

x2

x3

Gω

x

↑

x1

(a) T1

Gβ

G

c a

d

b

(b)

x→

Figure 6.1. (a) Schematic equilibrium phase diagram of binary alloys (Ti–V, Zr–Nb) from Groups 4 (solvent) and 5 (solute) (marked by solid lines). Superimposed metastable phase boundaries have also been shown by broken lines. Cross-over between Ms () and Ms () curves mark the composition range beyond which the -phase forms in preference to the martensites ( ) phase upon quenching (shaded region in the figure). (b) Schematic free energy curves for the -, - and the -phases. The letters ‘a’ and ‘b’ shown in the figures correspond to the compositions of and in equilibrium; ‘c’ and ‘d’ correspond to the compositions of and in metastable equilibrium and ‘e’ and ‘f’ correspond to Ms ( ), Ms (), respectively.

common tangents which represent the equilibria (stable or metastable) between (i) the - and the -phases and (ii) the - and the -phases defines the compositions corresponding to the end points of the tie lines within the + and the + phase fields. The composition limits for the formation of the -phase can be indicated with reference to the schematic phase diagram shown in Figure 6.1(a). The athermal → transformation occurs in alloys with compositions in the range x1 <x<x2 , while aged forms on isothermal treatments in the composition range x1 <x<x3 .

Transformations Related to Omega Structures

477

In general, the martensitic or phase does not coexist with the -phase, although there are some recent reports which show the presence of -particles in the retained -regions coexisting with martensitic -plates. The competition between the martensitic → and the → transformations has been studied in Ti–Nb alloys as a function of the alloy composition and of the severity of quench (Moffat and Larbalesiter 1988). They have found that the martensitic -phase forms in alloys containing upto 25 at.% Nb when the quenching rate is in excess of 300 K/s. Under slower quenching rates (∼0.3–3 K/s) the -phase precipitates in alloys containing up to 50 at.% Nb. Only alloys containing 60 at.% or more Nb have been found to retain the -phase fully upon quenching. While the lower concentration limit (25 at.% Nb) defines the point where the Ms ( ) curve interests the Ms () curve, the higher concentration limit (60 at.% Nb) corresponds to the point where the boundary of the metastable ( + ) phase field approaches room temperature. The Zr–Nb alloy system has been studied in detail with reference to the -transformation. The Ms ( ) and the Ms () curves in this system intersect at a Nb concentration of about 7 at.%, the Ms () curve comes down to room temperature at nearly 18 at.% Nb and correspondingly the metastable boundary of the ( + ) phase field approaches room temperature at a concentration level of about 30 at.% Nb (Figure 6.2). Table 6.1 shows the composition limits within which athermal and thermally activated -formation occurs in several alloy systems. The composition x( /) at which the Ms ( ) curve intersects the Ms () curve in a binary alloy system can be estimated from the concentrations corresponding to the boundaries

1136 K

β

Temperature (K)

α+β

β1 + β2

893 K

Ms(α ′)

18.5 β +ω 673

Ms(ω)

α′ 0

Zr

β +ω 10

1073

298

Metastable β 20

30

Atomic % Nb

Figure 6.2. Binary phase diagram of the Zr–Nb system showing the superimposed Ms () and Ms () lines. Composition ranges in which martensitic , athermal in the -matrix and metastable -structure are formed on -quenching are indicated.

478


Table 6.1. Comparison of calculated and observed compositions where Ms ( ) and Ms () lines intersect in different Ti- and Zr-based alloys. Alloy system

Ti–V Ti–Cr Ti–Mn Ti–Fe Zr–Nb

Onset of athermal -formation (at.%) compositions at which Ms ( ) and Ms () intersect Calculated

Observed

16 8 6 5 9

14 6 4 4 7

Maximum solute content (at.%) of aged -formation observed

25–30 18–20

30–33

of two phase fields, either the equilibrium ( + ) field or the metastable ( + ) field. The slope of the To versus x plot, where To is the temperature at which the chemical free energies per unit volume of the parent and the product phases are equal and x denotes the composition (in at.% solute), can be related to the changes in the free energy of the solution G→ at xo and the changes in the entropy of the solvent, S → , due to the change in phase from to either or as:

→ dTo /dxx→ = Gx→ /STiZr o o

(6.1)

The above expression is based on the regular solution model. At low solute concentrations

→ Gx→ = RT lnx /x + STiZr T o

(6.2)

which reduces to

= RT lnx /x Gx→ o

(6.3)

The second term on the right hand side of Eq. (6.2) is much smaller compared to the first term. The simplifying assumptions and some data which are used for the estimation of the composition x ( /) at which the quenched product changes from the to the -structure in binary alloys of Ti and Zr are listed below: (a) The heats of transformation for the → transformation in Ti and Zr are taken from Kubaschewski et al. (1993) as 4.2 kJ/g mole for Ti and 3.9 kJ/mole for Zr which yield the values of S → as 0.74 for Ti and 0.85 for Zr.


479

(b) The steep slopes of the / transus in the pressure–temperature phase diagrams of pure Ti and Zr indicate that the differences in entropy between the - and the -phases in these metals are very small. In view of this S → is taken to be approximately equal to S → . (c) The To versus x plot is assumed to be linear. (d) The composition at which the Ms curves for the → and the → transformations intersect is taken to be the same as the point of intersection of the respective To lines. This implies that the supercooling (To − Ms ) required for the two transformations is assumed to be equal. This approximation is valid as the extents of supercooling required for these transformations are quite small due to a relatively small contribution of the strain energy associated with the nucleation of either of the product phases in these systems. The composition, x( /) can then be expressed as

x / =

→ → − TTiZr TTiZr

dTo→ /dx − dTo→ /dx

(6.4)

The difference between the / and the / transition temperatures for pure Ti and Zr being about 400 K, one obtains by substituting Eq. (6.1) in Eq. (6.4)

− G→ x / = S → 400/G→ xo xo

(6.5)

Using the Eqs. (6.3) and (6.5) the composition, x( /), at which the quenched product changes from to has been calculated and the result has been compared with the experimental value, wherever available (Table 6.1). 6.2.2 Formation of equilibrium -phase under high pressures High-pressure resistivity studies on pure Zr first showed a phase transition at a hydrostatic pressure of about 6.0 GPa (Bridgman 1948a,b). Subsequently, Jayaraman et al. (1963) detected transformations in Ti at 8.0–9.0 GPa and in Zr at 5.0–6.0 GPa using the same technique. The transformation was initially thought to be the → transformation, as the -phase was supposed to be the denser phase at that time. Jamison (1963) was the first to identify the high-pressure phase to be one with a simple hexagonal structure (-phase). Jamison also pointed out the difficulties in determining the equilibrium / phase boundary in Ti and Zr, as a large hysteresis is associated with the transformation. It has been observed (Jamison 1964) that the -phase, once formed by the application of a high pressure, does not revert back fully to the -phase on removal of the pressure. The metastable

480


retained -phase remains in association with the equilibrium -phase at ambient pressure in samples which have been subjected to a pressure treatment. A complete transformation of the -phase into the -phase requires heating Ti to 380 K and Zr to 470 K for several hours after the removal of pressure. Using the equilibrium transformation data compiled by Kutsar (1975) for the → , → and → transformations, Sikka et al. (1982) have constructed a schematic pressure–temperature phase diagram (Figure 6.3) for the Group 4 transition metals. The triple point coordinates (Pt , Tt ), the equilibrium → transformation temperature at 0.1 MPa, T ( → ), the equilibrium / transformation pressure at 293 K, Po (/), and the → reversion temperature at 0.1 MPa, T ( → ), for Ti, Zr and Hf are indicated in Table 6.2. The hysteresis observed in the → transition is represented in Figure 6.3 by a pair of broken lines corresponding to the completion of the → and the → transitions. The fact that

Equilibrium lines Hysteresis lines

Temperature (K)

α

β

β

T

(Pt , Tt) α ω

ω

α α

T

ω

293

α

ω

Poα – ω

Pressure (GPa)

Figure 6.3. Schematic pressure–temperature phase diagram of Group 4 transition metals. Equilibrium lines are shown by solid lines and the values of the triple points are given in Table 6.2 for all the Group 4 metals. The broken line represents the hysteresis in the transformation from one phase to another. This hysteresis could be due to the presence of interstitial solutes in the pure metals.

Table 6.2. Calculated values of triple points for the Group 4 transition metals.

Pt (GPa) Tt (K) T ↔ (K) P ↔ (GPa) T ↔ (K)

Ti

Zr

Hf

9 1100 1155 2 380

6 975 1135 2.2 470

30 1800 2030 21 –


481

Table 6.3. Calculated values of the thermodynamic parameters required for various transformations in Group 4 transition metals. Ti

Zr

Hf

→

→

→

→

→

→

→

→

→

V (cm3 /mol)

−0032 −0036 −0052

0.18

0.14

−009

−029 −019 −02

0.1

−016 +01 −002

−045

0.43

S (J/mol/K)

3.68 3.64

−159

5.18

3.55

−125

4.81

3.34 2.51

2.93

5.85

dT /dp (K/GPa)

8 −26 −9 −10

90

−24

160

38 −7

150

70

27 112

−26

21

the -phase is partially retained at ambient conditions after a pressure treatment has also been depicted in this figure as the → hysteresis line intersects the temperature axis at 0.1 MPa at a temperature higher than room temperature. Additional data regarding the volume and entropy changes associated with different phase transitions and the slopes of various phase boundaries in the P–T plane are listed in Table 6.3. The → transition has also been studied in dynamic (shock) experiments involving pressure pulses of microseconds duration. Phase transitions under such conditions are usually revealed in the observed discontinuities in the plots of shock velocity (Us ) versus particle velocity (Up ). McQueen et al. (1970) have noticed the discontinuities in the Us versus Up plots for Ti (17.5 GPa, 370 K), Zr (26.0 GPa, 540 K) and Hf (40.0 GPa, 725 K). The transition pressures under shock loading conditions have been found to be substantially higher than those reported for static pressure investigations. An extensive hysteresis in the / transition has also been noticed in shock pressure experiments. 6.2.3 Combined effect of alloying elements and pressure in inducing -transition As pointed out in the preceding two sections, the -phase can be generated in Ti, Zr and Hf systems either by the addition of adequate amounts of a stabilizing alloying element or by the application of pressures sufficient for inducing the -transition. The combined effect of alloying elements and pressure has been studied by only a few investigators (Afronikova et al. 1973, Ming et al. 1980, Vohra et al. 1981, Dey et al. 2004). High pressure studies on Zr–Nb,

482


the Ti–Nb and the Ti–V alloy systems have brought out the following common features: (a) The application of pressure promotes the formation of the -phase in the composition range where the -quenched structure is martensitic ( ). The composition range over which the -phase is present is, therefore, wider under pressure. (b) The -start pressure is lowered with increasing concentration of a -stabilizing element up to a certain point beyond which the -start pressure rises with further increase in solute concentration. The results of high-pressure studies on Ti–V alloys up to a pressure of 25.0 GPa are shown in the pressure–composition phase diagram in Figure 6.4. It may be noted that the boundaries of the different phase fields do not correspond to the equilibrium condition as these results are compiled from X-ray diffraction and resistivity data which were obtained from samples maintained under pressure, whereas transmission electron microscopy results pertain to pressure-treated samples which were brought back to ambient conditions. The features of the pressure–composition phase diagram in Figure 6.4 could be rationalized in terms of a set of hypothetical free energy versus concentration plots, as shown in Figure 6.5. Thus the essential features of experimental observations, such as the extension of the composition range of -formation and the variation in the -start pressure with concentration, could be explained on the basis of these

Pressure, GPa

20

ω

ω+β β

10

α+ω α Ti

10

20

30

40

50

V, Atomic %

Figure 6.4. Pressure–composition phase diagram of the Ti–V system at room temperature. As could be noticed from the figure, addition of V in Ti reduces the pressure for → phase transformation. Upon comparing this figure with Figure 6.2, a similarity between the trends in the transition temperature and transition pressure could be noticed.

Transformations Related to Omega Structures α

ω

α

α A

P = Po

P = P1 T = 300 K

β

↑ F

β

α+β

α α+ω

Composition →

α

ω

T = 300 K

β

↑ F

483

B

β

β+ω

Composition →

A

P = P2

α

T = 300 K

B

P = P3 T = 300 K

ω β ↑ F

α

A

α+ω ω

ω+β

Composition →

ω

↑ F

β

β

ω B

A

β ω+β Composition →

B

Figure 6.5. Hypothetical free energy curves as a function of composition and pressure. These curves are drawn based on the phases observed at various pressures and alloy compositions.

hypothetical free energy–concentration plots. Under the equilibrium condition a system possessing such concentration dependence of free energies for three competing phases would be expected to exhibit an eutectoid phase reaction, as shown in Figure 6.6. Though experimental confirmation of an eutectoid phase reaction in a -forming system in the pressure–concentration plane is still not available, the observed metastable Ti–V phase diagram (Figure 6.4) and a generalized version of the same (Figure 6.6) bring out the basic trends in relative phase stabilities as functions of pressure and concentration. Of late the formation of the -phase was observed in alloys upon shock loading (Hsiung and Lassila, 2000). These alloys under normal conditions do not show the formation of the -phase. Recently, Dey et al. (2004) have shown presence of in the Zr–20 Nb alloy upon shock loading. The morphology of the -phase has exhibited plate-like morphology. In both these studies the plane of contact between the - and the -phases was the

484


P3

ω

Pressure

↑ P2

β+ω

α+ω

P1

α P0

Composition

↑

A

β α+β

B

Figure 6.6. A generalized version of the phase diagram constructed on the basis of the free energy plots shown in Figure 6.5. This phase diagram suggests the presence of an eutectoid phase reaction in the pressure–composition phase diagram.

{112} plane of the -lattice. Such observations showed that the -structure can also be visualized as a defect in the bcc lattice akin to twinning (Hsiung and Lassila, 2000).

6.3

CRYSTALLOGRAPHY

The crystallography of the → transformation has provided the essential clue to the transformation mechanism and is, therefore, a subject which has been investigated in great detail. Soon after the -phase was first reported, a controversy arose regarding the crystal structure of this phase. The structure of the -phase is described in the following section in which a summary of the early controversy is also given. The lattice correspondences between (i) - and -phases and (ii) - and -phases are discussed subsequently. 6.3.1 The structure of the -phase There has been a serious controversy regarding the crystal structure of the -phase. Early studies suggested that the -phase had a complex bcc unit cell with a lattice parameter three times that of the -phase. In fact, selected area diffraction (SAD) patterns taken from the + structure corresponding to all zones excepting the


485

and the zones, invariably show extra diffraction spots which divide all bcc reciprocal lattice vectors into three equal segments. It has subsequently been recognized that the occurrence of all the variants of -particles which are distributed in the -matrix gives rise to composite diffraction patterns which possess the symmetry of the matrix. The -structure has later been determined to be either hexagonal (belonging to the space group D16H (P6/mmm) (Silcock et al. 1955) or trigonal (belonging to the ¯ space group D33D (P3m1) (Bagaryatskiy et al. 1955). Both these structures can be generated from the parent bcc structure by simple atom movements, as discussed in the following section. The discrepancy between the reported hexagonal and trigonal structures of the -phase has also been resolved in a study carried out by Sass and Borie (1972) who have shown that increasing solute (-stabilizer) concentration causes the sixfold symmetry characteristic of D16H to degenerate into the threefold symmetry characteristic of D33D . 6.3.2 The – lattice correspondence The lattice correspondence between the - and the -structures not only establishes the crystal structure of the -phase but also provides a clue to the transformation mechanism. The orientation relationship between the two phases has been determined in a large number of investigations and has been unanimously described as ¯ > < 1120 ¯ > 111 0001 < 110 The relation between the lattice geometries of the two phases can be best illustrated in a diagram showing the stacking sequence of the (222) atomic planes of the bcc structure (Figure 6.7(a)). The ABCABC stacking sequence of these atomic planes generate the open bcc structure with the threefold axes along the direction. The -lattice can be created by collapsing a pair of (222) planes to the intermediate position, leaving the next plane undistorted, and collapsing the next pair of planes and so on (Figure 6.7(b)). When the collapse is complete a sixfold rotation symmetry is created around the specific direction along which lattice collapse occurs (Figure 6.7(c)). In case the collapse is incomplete, the trigonal symmetry is not lost, and the resulting -structure is associated with a trigonal structure with the space group D33D . The - and the trigonal structures can be formed by introducing a displacement wave in the parent bcc () structure. This point is illustrated in Figure 6.7(d) where a displacement wave with the wave vector, K= 2/3 < 111 >, can create the -structure when the amplitude of the wave is sufficient for moving {222} planes by a distance 1/2{222}. When the amplification of the wave is smaller, the resulting structure is trigonal. On the basis of the lattice correspondence, shown in Figure 6.7, the lattice parameters of

486

Phase Transformations: Titanium and Zirconium Alloys [111]

[0001] Layer No. 3 2 1.5 1 0

[011]β [110]β

(222)

[101]β

[1210]

(0001)

[2110]

ω-hexagonal

β bcc (a)

(b)

2 Layer 1 Layer 0 Layer [2110] (c)

Displacement Up

[1210]

+ – 0

1

2

3

4

5

6

(d)

Figure 6.7. Planar arrangement of the {222} atomic planes in the bcc lattice; (a) the ABCABC sequences of atomic plane arrangements could be noticed (planes are marked as 0,1,2,3 ). (b) The planar arrangement of the (0001) atomic planes in the -lattice. As could be observed the -lattice could be obtained from the bcc lattice by collapsing a pair of planes (1 and 2, as shown in figure) in the middle (marked as 1.5) while keeping the third one (0 and 3) undisturbed. (c) Completely collapsed {222} planes form a sixfold symmetry whereas an incomplete collapse forms a threefold symmetry along the axis. (d) The displacement of {222} planes could also be visualized in terms of a sinusoidal displacement wave where an upward movement is seen as a positive displacement and a downward movement as a negative displacement.

the -structure (a and c√ ) can be expressed in terms of the bcc lattice parameter, √ a :a = 2 a and c = 3/2.a . A comparison of the lattice parameters of the -phase computed from the a value of the corresponding -phase and those experimentally observed shows that the divergence between the two is negligibly small for the athermal → transformation (Table 6.4). A similar comparison between the corresponding lattice dimensions of the -phase and the aged -phase, which is in metastable equilibrium, reveals that the isothermal → transition is associated with a linear contraction of about 5% (volume contraction ∼15%). Since there are four sets of directions, there are four possible crystallographic variants of the -structure in a given bcc parent crystal. The matrix representation of the lattice correspondence for all the four variants is shown in


487

Table 6.4. The calculated and experimentally determined values of the lattice parameters of the -phase encountered in different alloy systems. Alloy system

Calculated values of lattice parameters (nm)

Experimentally determined values of lattice parameters (nm)

Difference (per cent)

Zr–Nb

a = 0.5003 c = 0.3064

a = 0.5019 c = 0.3089

a = 031 c = 082

Zr–Nb–Cu

a = 0.5003 c = 0.3064

a = 0.5029 c = 0.309

a = 052 c = 085

Ti–Nb

a = 0.4596 c = 0.3031

a = 0.4627 c = 0.2836

a = 067 c = 643

Ti–V

a = 0.4596 c = 0.28146

a = 0.460 c = 0.282

a = 008 c = 019

Table 6.5. Correspondence matrices relating the -(bcc) and the -(hexagonal) structures. Variant


1

¯ [12 ¯ 10] ¯ (111) (0001) ; [110]

2

¯ [12 ¯ 10] ¯ ¯ (0001) ; [1¯ 10] (111)

3

¯ [12 ¯ 10] ¯ ¯ (0001) ; [110] (1¯ 11)

4

¯ 10] ¯ ¯ (0001) ; [110] [12 (111)

Correspondence ⎡ 2 2¯ ⎣0 2 2¯ 0 ⎡ 0 2¯ ⎣ 2 2¯ 2¯ 0 ⎡ 2¯ 2 ⎣ 0 2¯ 2¯ 0 ⎡ 0 2 ⎣ 2¯ 2 2¯ 0

matrix [R] ⎤ 1 1⎦ 1 ⎤ 1¯ 1⎦ 1 ⎤ 1¯ 1¯ ⎦ 1 ⎤ 1 1¯ ⎦ 1

Table 6.5. Within the same orientation variant of the -structure, associated with a given variant of lattice collapse, three distinct subvariants can be created, depending upon whether the A, B or C plane remains as the uncollapsed plane. These subvariants are schematically illustrated in Figure 6.8. An -region within the bcc phase remains coherent with the parent, as seen from the small difference in their lattice dimensions. However, given the total number of variants and subvariants (4 × 3 = 12), the elastic strains (arising from the atomic displacements and the slight change in volume) associated with different particles interact with one another. This elastic interaction is a controlling factor which governs the arrangement of -particles in the -matrix.

488

Phase Transformations: Titanium and Zirconium Alloys ω -Subvariants A B C A B C A B C A

A

B

C

Uncollapsed plane

Figure 6.8. Schematic presentation of the translational subvariants of the -phase produced due to the random collapse of different {222} planes of the bcc lattice. Regions of the different subvariants are marked in the figure.

6.3.3 The – lattice correspondence Diffraction experiments have established that the following two orientation relations exist between the - and the -phases: Orientation Relation I (OR I) ¯ ¯ ¯ 0001 0111 Orientation Relation II (OR II) ¯ ¯ 0001 0001 1120 On the basis of these observed orientation relations two distinct mechanisms, involving either the direct → route or the two stage → → route, have been proposed for the transformation from the to the -phase. Silcock et al. (1955) first proposed a model for the formation of the -lattice directly from the lattice which leads to the orientation relation described in OR II. The shift in atomic positions required for the direct → transformation is shown in Figure 6.9. The (112¯ 0) plane is generated from the (0001) plane by displacement of alternate hexagons on the (0001) plane by 0.148 nm (for Ti). The atoms marked a, b, c and d shift to positions a , b , c and d . In addition, a contraction of ∼4.7% along ¯ directions are required ¯ and an expansion of ∼4.5% along the [1100] the [1120] for the generation of the -unit cell. Usikov and Zilbershtein (1973) have suggested that the → transformation occurs via the intermediate -structure which is unstable at the pressures and temperatures under consideration. This suggestion is based on the fact that the


489

0.7667 nm

d′

0.295 nm

d

c′

a′

c

a

b′

b

[1120]α ⎥⎪ [0001]ω [1100]α ⎥⎪[1100]ω (0001)α⎥⎪(1120)ω

¯ plane of the Figure 6.9. Lattice correspondence between the - and the -phases. The (1120) -phase is generated from the (0001) plane of the -phase by shifting the atoms at a, b, c, d to a , ¯ and expansion along [1100] ¯ direction is also needed b , c , d positions. Contraction along [1120] to generate the -unit cell.

orientation relation observed by them (OR I) can be described in terms of the product of the correspondence matrices R and R which pertain to the → and the → transformations, respectively. They have shown that all the observed orientation relations between the - and the -phases, as revealed from diffraction patterns, are consistent with the correspondence matrix associated with OR I. Later TEM studies (Vohra et al. 1980, 1981) on high-pressure treated Ti–V alloys, subjected to high pressures where alloying addition promotes -phase formation, have confirmed the presence of the -phase along with the -phase in the matrix of -phase grains. This observation provides a direct evidence of the occurrence of the -phase as an intermediate state during the → transition. The fact that the stability of the -phase is enhanced by V additions appears to have been responsible for the retention of this intermediate phase. Orientation relations have also been determined in samples in which the -phase has been made to form by the application of shock pressure. Kutsar et al. (1990) have reported that the / orientation relation in shock-treated Zr matches with OR II. In some recent investigations Song and Grey (1994, 1995) have observed a new orientation relation, as described below, between the - and the shock-induced -phases: ¯ 1100 ¯ ¯ 0001 1011 On the basis of this observation they have proposed a different transformation mechanism according to which the -phase forms directly from the -phase. Jyoti

490


]

d 1120 [

j

e

k 10

]

12 0°

(2.5725)

[10

f

c

(2.5725)

h

j

(2.5725)

g k

(2.5725)

e

(2.5418)

f

i b

d

a

(0001)α (a)

b

(2.5418)

j

e (+ 4219)

k

h

(2.1098)

f h

(– 4219)

a

i

d [11

g

(2.5418)

(2.5725)

]

01

3] 12 c

[1

(2.5418)

109 °

0] 10 c [1

(2.9533)

b

(011)β (b)

a

i

(1011)ω (c)

Figure 6.10. Schematic presentation of (a) the (0001) plane of the -structure (b) the (011) plane ¯ plane of the structure. Filled circles represent atoms on the plane of the and (c) the (1011) whereas open circles represents atoms away from the planes. The distance of separation is given in parenthesis (in Å). A reference set of atoms is marked on each projection. As could be noticed very small movement of atoms j, k, h and g on (0001) plane is needed to obtain either the -planar or the -phase. A close resemblance suggests a possible path of the → phase transformation may have the -phase as the metastable phase as an intermediate path. Results are more inclined to direct → transformation (for detail see text).

et al. (1997) have examined SAD patterns belonging to a larger number of zone axes for determining the orientation relation between the - and the -phases and have attempted a harmonization of all the conflicting results reported earlier and have demonstrated that nearly all the orientation relations reported earlier can be expressed as subsets of OR I. They have considered the possibility of the formation of the -phase through an intermediate -structure and have compared the crystallographic results with those expected from the direct formation of the -phase from the -phase. A schematic representation of the atomic arrangements pertinent to either of the routes is shown in Figure 6.10. Small variation in the positions of atoms marked in the figure generates all the three structures indicating a close resemblance among these structures.

6.4

KINETICS OF THE → TRANSFORMATION

The → transformation has been found to proceed under two different conditions, namely, during quenching from the -phase field and during isothermal holding at a temperature below about 773 K. The kinetic characteristics of these two cases correspond to an athermal and a thermally activated process, respectively. A comparison between these two modes of -phase formation is made in the following two sections.


491

6.4.1 Athermal → transition On quenching an -forming alloy from the -phase field to a temperature below Ms (), the athermal → transformation can be initiated. The initiation of the athermal -phase is characterized by diffuse intensity distribution in diffraction patterns, as is illustrated in Figure 6.11. The athermal nature of the transformation has been established through ultrahigh quench rate (∼11 000 K/s) experiments (Bagaryatskiy and Nosova 1962). The facts that the transformation is not

00021

110

00011 000

(a)

002

(b)

(c)

Figure 6.11. (a) Diffuse intensity patterns obtained on the selected area diffraction patterns from the of the bcc lattice. (b) Schematic drawing exhibiting deviation from the 1/3, 2/3 position from the rel vector [112]. (c) Diffuse intensity distributions in various zones.

492


suppressible even under such quenching conditions and that it occurs at temperatures where the diffusivity of both self and solute atoms is negligibly small suggest that the transformation can occur without any thermal activation. The diffuse intensity distribution gradually changes to sharp -reflections as the temperature is progressively lowered below Ms (), suggesting an increase in the volume fraction of the -phase and a growth of individual -particles. The complete reversibility of the transformation is yet another evidence for its athermal nature. All these observations suggest that this transformation can be classified as a displacive transformation. Particles of the -phase that form on quenching do not exhibit a plate-like morphology or a surface relief effect which are both characteristic features of a martensitic transformation. It is for these reasons that the athermal → transformation is not categorized as a martensitic transformation in spite of its athermal and composition-invariant character. 6.4.2 Thermally activated precipitation of the -phase Precipitates of the -phase form in a -phase matrix on ageing at temperatures below about 773 K, which is the upper cut-off temperature for -phase formation by ageing in most of the -solid solutions. The progressive increase in the volume fraction and the growth of -particles with an increase in the isothermal holding time are indicative of a thermally activated transformation mechanism. Time-temperature-transformation (T-T-T) plots pertaining to isothermal -precipitation have been experimentally generated in several alloys. Figure 6.12 shows two such plots corresponding to Zr–Nb alloys. The characteristic C-shape of these T-T-T plots is consistent with the well-known dependence of the incubation period on the driving force and the diffusivity which change in opposite directions with a lowering of temperature. An in-situ study of -precipitation under 1 MeV electron irradiation in a highvoltage electron microscope has revealed the real time changes (Figure 6.13) in diffraction patterns and has demonstrated the progressive sharpening of -reflections from the initial diffuse intensity distribution. This observation is remarkably similar to the evolution of sharp -reflections from the diffuse intensity distribution which has been reported to occur in the athermal → transformation as the temperature is progressively lowered. The lattice correspondence between the - and the “aged ” structures has also been found to be the same as that recorded in the case of the athermal → transformation. These features suggest that isothermal -precipitation occurs by an atomistic mechanism which is essentially the same as that operative in the athermal → transformation. The thermally activated component of the overall transformation mainly involves a solute-partitioning process which accompanies the lattice collapse mechanism

Transformations Related to Omega Structures 900

α + β Zr + βNb α BY DILATOMETRY

Temperature ~°C

β 800

ω + β Zr α + β Zr

700

β/α + β Zr

600

493

α + β/α + β Nb

α + β Zr

α + β Zr + βNb

500

400

β

ω + β Zr

300

200 100

ωr 0.2

1h 0.5

1

2

5

10

102

5

2

1 day

6h 2

5

103

4 days

2

5

104

Time ~ Min

(a) 900 800

β ω + βZr α + βNb+ β βI/βI + βII

700

Temperature ~°C

α + β Nb + β Zr α + βNb

β + βI + βII

βI + βII/α + βNb 600

β + βNb + β

500

β

α + βNb + β Zr

α + βNb

α + β Zr

400

ω + β Zr 300 200

1h 100

0.2

0.5

1.0

2

5

10

2

5

6h 10

2

4 days

1 day 5

10

2

5

10

Time ~ Min

(b)

Figure 6.12. T-T-T diagrams for (a) Zr–12% Nb alloys and (b) Zr–17.5% Nb alloys. Increasing Nb concentration retards the formation of the -phase from 3 h at 300 C for Zr–12% Nb to 2 days at 300 C for Zr–17.5% Nb alloy.

494

Phase Transformations: Titanium and Zirconium Alloys 300 K

t =0 s 450 K

t=0 s

60 s

300 s

120 s

240 s

Figure 6.13. Selected area diffraction patterns from the -phase regions after irradiating with 1 MeV electrons for different time periods. Gradual transformation of the diffuse intensity to the -reflections could be noticed.

(as discussed in a later section). It may be noted from Figure 6.13 that the -transformation can occur even at ambient temperature due to the condition of enhanced diffusion obtained under 1 MeV electron irradiation. 6.4.3 Pressure-induced → transformation It has been noticed that the transformation pressure, Ps→ , determined by different research groups, shows a fairly high scatter for pure Ti (from 2.8 to 9.0 GPa) and for pure Zr (2.2 to 6.0 GPa) (Sikka et al. 1982). This large scatter has been attributed to variations in the pressure exposure time, the starting microstructure, the impurity content and the hydrostatic component of the applied pressure. The influence of the exposure time at a constant pressure at ambient temperature on the → transition has been systematically studied and the time dependence of the growth of the -volume fraction has been established. The rate of the -transformation as a function of pressure shows a peak (Figure 6.14) (Zilbershtein et al. 1975, Vohra 1978) analogous to that observed in nucleation rate versus temperature plots pertaining to thermally activated martensitic transformations (isobaric). The time-dependent nature of the pressure-induced → transformation suggests that there exists a barrier, possibly that of the nucleation step, which can be overcome by thermal activation. The nucleation step involves the growth of quasistatic -embryos to the critical size by thermally assisted diffusion processes. After attaining the critical size, the nuclei grow spontaneously in an athermal manner. The observed peak in the rate-versus-pressure plot is consistent with the fact that the driving force increases while the diffusivity decreases with increasing pressure.

Transformations Related to Omega Structures x x 3

x

x

x

x

2

x

xx x x

x

1

0

2.0

xx

ω

6.0

10.0 2.0

x

x

xx xx α ω ↑

x x x x x x P Mα S

↑

Rate of transformation arbitrary unit

4

495

P MS 6.0

xx

x x 10.0

P (GPa) (a)

(b)

Figure 6.14. The rate of -phase transformation with pressure shows peaking in between (a) 6–7 GPa in the case of Ti and (b) in between 5–6 GPa in the case of Zr. Theoretically calculated start pressures for the → transformation are also shown.

6.5

DIFFUSE SCATTERING

The -forming systems, in certain composition ranges, invariably exhibit diffuse intensity distribution in electron, X-ray and neutron diffraction patterns. The complexity of these patterns, though far from being fully understood, has induced many research groups to study this unusual displacive phase transformation. The essential features of the experimental observations regarding the diffuse intensity distribution and the soft phonon behaviour associated with the -transformation are summarized here. Diffracted intensity from the athermal -phase is often diffuse, centred away from the “ideal ” positions in the reciprocal space and can show pronounced curvature and asymmetry. These effects are predominantly manifested as the alloy system is moved away from the region of relative -stability in the phase diagram along either the concentration or the temperature axis. The nature of the diffuse intensity in electron diffraction patterns is shown in Figure 6.11. The diffraction pattern corresponding to the [110] zone axis (Williams et al. 1973) clearly shows that the intensity maxima in the diffuse intensity distribution are located at positions √ slightly away from those expected for the ideal -structure (c = 3/2.a and a = √ 2.a , with the lattice correspondence described earlier). For the ideal -structure the (0001) and the (0002) reflections should occur at 1/3 and 2/3 of the distance between the (000) and the (222) reflections, respectively. The line drawn along the direction through bcc spots in Figure 6.11 shows where the reflections corresponding to the ideal -structure should be located. The observed (0001) reflection is displaced from the ideal position away from the (000) spot, while the

496


(0002) reflection is displaced towards the (000) spot. Dawson and Sass (1970) have shown that the extent of this deviation depends on the stability of the -phase with respect to the -phase which, in turn, is decided by the alloy composition and the departure from the Ms () temperature. The observed diffuse intensity in electron diffraction patterns corresponding to [110] and [102] zone axes appears in the form of quasicircular arcs (de Fontaine et al. 1971). In-situ experiments carried out using a cooling stage in an electron microscope have shown that the diffuse intensity gradually transforms into sharp -reflections as the sample is cooled to temperatures sufficiently below the Ms () temperature. Reheating of samples has shown a gradual disappearance of the -reflections and the reappearance of diffuse patterns. The complete reversibility of the process has been demonstrated by the experiments of de Fontaine et al. (1971). The possible occurrence of multiple scattering has been considered for explaining the appearance of diffuse streaks, especially the quasicircular arcs. Multiple scattering will no doubt alter the relative streak intensities, but it cannot account for the presence of the streaks themselves. Examining several zones of electron diffraction patterns and considering the symmetry of the reciprocal lattice, de Fontaine et al. (1971) have constructed a three dimensional model of the diffuse intensity (Figure 6.15) which is distributed on quasispherical surfaces centred around the 002

020

000

110

200

Figure 6.15. Three-dimensional model of the intensity distribution of the diffuse intensity in the reciprocal space of the bcc unit cell. The sphere of diffuse intensity touches the octahedra of {111} faces which surrounds it.


497

octahedral sites 100, 111, , of the fcc lattice (reciprocal of the bcc -lattice). These spheres of intensity touch all the {111} faces of the octahedra surrounding them. Below the Ms () temperature, -reflections appear in the form of discs lying on {111}∗ reciprocal lattice planes at the points of tangency of the intensity spheres with the faces of the octahedra. The coordinates of these points are 1/3 , 2/3 etc., which can be identified as -reflections. With lowering of temperature the spherical surfaces on which the intensity is distributed become the octahedra themselves which are seen as streaks along traces of {111} planes. Neutron diffraction experiments (Sikka et al. 1982) have also shown that with increasing solute content, the -reflections become diffuse and are shifted away from ideal -positions in the reciprocal space. Figure 6.16, which shows the (011) reciprocal lattice section for a Zr–20% Nb alloy, clearly demonstrates this point. The shifts correspond to an increase in the -wave vector from q = 2/a )

h′ 3 0.0

4.0

4 8 12

1.0

2.0

3.0 4.0

3.0

3.0

h3

2.0

h′ 1

2.0

1.0

1.0

[111]β

0.0

↑

↑

0.0

–3.0

–2.0

–1.0

0.0

[211]β 1.0

2.0

3.0

h1

Figure 6.16. Neutron diffraction intensity plots showing distribution of the diffuse intensity on the (011) reciprocal lattice section of the Zr–20% Nb alloy (Keating and Laplace 1974).

498


(2/3,2/3,2/3) to q + q. The magnitude of q, which is a measure of the extent of incommensuration with respect to the ideal -lattice, is seen to be dependent on the solute concentration and the extent of undercooling below the Ms () temperature. However, in some systems, cooling to a temperature as low as 5 K does not change the diffuse intensity, suggesting the presence of the incommensurate structure even at such low temperatures. It has been noticed that with the application of pressure the -reflections get sharper and the structure tends to become commensurate. It has been noticed that longitudinal phonons having a wave vector K which is close to that for an “” phonon (K = 1/3 [222]) exhibit a sharp central peak flanked by a pair of weak diffuse peaks. The diffuse peaks are representative of the dynamic phonon structure whereas the central peak represents a static phonon. This observation can be interpreted in terms of a mechanism involving the propagation of a large amplitude dynamic phonon. As it propagates, it structurally degenerates into two alternate configurations, and anti- (described in Section 6.6) which have a large difference in free energy (Cook 1973). Large amplitude phonons are thus “pinned” in a metastable state at the low free energy -positions. The lattice collapse mechanism which is implicit in this interpretation will be discussed in detail in Section 6.6. The experimental longitudinal (L) and transverse (T) phonon dispersion curves for -Zr are shown in Figure 6.17 (Stassis et al. 1978). The pronounced dip in the longitudinal [111] branch at the reduced wave vector, m (= Km d/) 0.7, which is slightly away from the ideal , is indicative of a tendency of this mode Γ

N

H

Γ

P

25

25

[110]

[001]

[111] 20

Energy (meV)

20 L

L

L

15

15 T

10

10

bcc Zr T = 1423 K

5

0.5

0.3

0.1 0

0.2

0.6

T 5

1.0

0.8

0.6

0.4

0.2

0

Reduced wave vector

Figure 6.17. Experimental phonon dispersion curves for bcc Zr. A pronounced dip in the longitudinal [111] branch (marked as L) at a wave vector ∼0.7 could be noticed. This wave vector is slightly away from the wave vector needed for ideal which is 0.66 (Stassis et al. 1978).


499

to soften up. It is to be noted that the softening is not due to the extent that the system becomes unstable with respect to the development of this longitudinal displacement wave. Stassis et al. (1978) have also observed that the intensity of diffuse peaks as well as the extent of their shifts from the ideal -positions are temperature dependent.

6.6

MECHANISMS OF -TRANSFORMATIONS

As discussed in the preceding sections, the -phase can form in Ti and Zr base alloys in the following situations: (a) on application of hydrostatic pressure to the -phase – either by a static pressure or by a shock pressure (b) on quenching from the -phase field (c) on ageing the metastable -phase (d) on application of shock to the -phase (in -stabilized alloys such as Zr–20% Nb) The mechanisms involved in -phase formation in these cases are not the same. However, the lattice collapse mechanism, which converts the bcc structure to the hexagonal -structure, appears to be a common feature in all these cases and is, therefore, discussed first. 6.6.1 Lattice collapse mechanism for the → transformation The → transformation has been studied in detail and well documented in literature. The key experimental observations of these studies are summarized in the following: (1) The → transformation exhibits strong pretransition effects in terms of profuse diffuse scattering (e.g. Figure 6.11) and lattice softening (Sass 1972). (2) The → transformation cannot be suppressed by quenching. In the -quenched alloys the resultant -phase always remains finely divided typically less than 5 nm in diameter. Generally the shape of these -particles is ellipsoidal. All the possible four variants (corresponding to four directions) of -precipitates remain equally distributed in the -matrix. High-resolution electron microscopy (HREM) has confirmed the presence of all variants. (3) The -particles grow on ageing and during their growth, they gradually assume a cuboidal morphology with the cube faces parallel to {100} planes. Isothermal

500


ageing can also produce ellipsoidal -phase particles in concentrated alloys. The latter process is usually preceded by phase separation in the bcc -phase, resulting in solute rich and solute depleted -regions. (4) Under application of shock deformation, -plates have been produced (Hsiung and Lassila 2000, Dey et al. 2004) in the -phase. The product -phase assumes a plate shape and the transformation is reported to have similarities with the martensitic transformation. (5) The → transformation, resulting from thermal ageing, is accompanied by a reduction in the specific volume (typically about 5% (Hickman 1969)). Based on the above experimental features, the resultant -phase can be classified in the following categories: (1) Fine ellipsoidal particles athermally produced on quenching. Morphology of the particles suggests that there is no volume change associated with the → transformation. (2) Cuboidal or ellipsoidal -particles (20–50 nm), produced during isothermal ageing by a transformation resulting from diffusional alloy partitioning followed by lattice collapse. The transformation is accompanied by a volume change. (3) Plate shaped resulting from the shock pressure induced → transformation. The lattice correspondence between the - and the -structures, as described in Section 6.3.2, clearly shows that the -structure can be created from the -structure by collapsing a pair of adjacent (222) planes (e.g. 1 and 2 in Figure 6.7(b)), leaving the “0” and “3” planes undisplaced. This results in a transition from the ABCABC stacking of the bcc structure (in Figure 6.7(a), these three layers are depicted as “0 1 2”.) into the AB AB stacking of the -structure (shown as “0 1.5 3” in Figure 6.7(b)), where B planes correspond to the collapsed position located midway between B and C planes. A view along the [111] direction shows how the threefold rotation symmetry changes to a sixfold symmetry where the collapse is complete (Figure 6.7(c)). The ordered sequence of the displacement of the {111} type planes (Figure 6.7(d)) which causes the → transition can be represented by a displacement wave with wavelength = 3 d222 , the corresponding wave vector, K , being equal to 2/3 ∗ . The development of the longitudinal displacement wave (Figure 6.7(d)) along the [111] direction can, therefore, formally describe the transformation. The merit of this description lies in the fact that it can be applied to partial collapse situations by varying the amplitude of the displacement wave, the amplitude being directly related to the order parameter, . The →


501

transition in Ti and Zr alloys which occurs on quenching from the -phase field has been found to be associated with the following attributes: (a) athermal character; (b) presence of pretransition diffuse intensity associated with structural fluctuations, the diffuse intensity peaks being slightly away from the perfect -peaks; (c) stability of a two phase microstructure comprising fine particles of the -phase distributed in a -phase matrix immediately below the Ms () temperature. The following mechanism of the athermal → transition, based on the work of de Fontaine (1970), de Fontaine et al. (1971) and Cook (1973), takes these essential features into account while developing a Landau theory of displacement ordering for this transition. As shown in Figure 6.7(d), the → transition can be described in terms of the introduction of a displacement wave with the wave vector K = 2/3 ∗ in the bcc structure. A planar lattice model, as depicted in Figure 6.18, is very

(222) planes i=0

1

2 3

4

5

6

(a)

bcc

(d)

(b)

ω

(e)

[111]

Up

+ –

(f) Anti-ω

1

2

3

4

5

6

0

1

2

3

4

5

6

+ Up

(c)

0

–

Figure 6.18. Two possibilities of the collapse of the {222} planes are shown: (a) represents arrangement of the {222} planes, viewed on edge, in the bcc lattice. One possibility of collapse is shown in (b), where a two-plane collapse structure is shown (planes 1 and 2, 4 and 5), while in (c) a three-plane collapse structure is shown (marked as 2, 3, 4). The two-plane collapse produces the -structure whereas the three-plane collapse produces an anti- structure. (d) A collapse region in the bcc lattice showing an -region marked by an ellipse. This schematic provides a clue for the ellipsoidal morphology of the -phase. (e) and (f) are the wave representations of the - and anti- structures. The final positions of the planes are shown by broken lines.

502


useful in visualizing the process. It can be seen from the illustrations in Figure 6.18 that the -structure is produced by the -wave while a wave with an equal but negative amplitude (hence, negative ), represented in Figure 6.18(f), produces the anti-omega (−) structure in which both B and C layers are displaced towards the A layer. Needless to say, the anti- configuration in which three planes come close together due to the collapse mechanism will be energetically unstable. The fact that the anti- configuration is not equivalent to the configuration both in structure and in free energy suggests that the → transition cannot be of a second (or higher) order. Cook (1973) has put forward this symmetry argument to justify the attribution of a first-order character to this transition which has been experimentally vindicated by the observation of the occurrence of the two phase + structure. Considering the planar lattice, the free energy change associated with the lattice collapse can be expressed in the form 1 1 1 G = ij ui uj + ijk ui uj uk + ijkl ui uj uk ul + 2 3 4

(6.6)

where ij , the interplanar coupling parameters, are defined by the appropriate derivatives of the free energy with respect to the planar displacements which are denoted by ui , uj and are respectively normal to the planes i, j , numbered in sequence from the original at i = 1. The coupling parameters in Eq. (6.6) are not independent but are linked by the energy translational and rotational invariance relations (Leibfried and Ludwig 1961). Substituting into Eq. (6.6) the deformation u1 = 0 u2 = −u3 = a 3/12 ui+3 = ui (6.7) where a is the lattice parameter and the Landau order parameter for the displacement ordering in the → transition, the free energy change can be expressed after rearrangements in terms of the standard Landau expression G = A2 + B3 + C4

(6.8)

Schematic free energy versus order parameter plots (Figure 6.19) can show the relative stabilities of the - and the -structures at different temperatures. As is expected in a first-order transition, the stability of the -structure is brought about by a negative value of the third-order constant (B) which is responsible for lowering the free energy below zero at T < To (). With reduction in temperature, the magnitude of the third-order constant (B) increases and as temperature approaches Tm , a metastable state appears at a positive . With further reduction in temperature, the potential well of the metastable state deepens and the minimum appears


503

T > Tm ΔF

T > Tm

To(ω) < T < Tm T = To(ω)

η T < To(ω)

Figure 6.19. Schematic presentation of the free energy of two ( and ) phases as a function of order parameter. The asymmetric nature of the free energy curves represents the first-order nature of the -phase transformation. At temperature Tm , a local minimum free energy appears which deepens as temperature of the system reduces in the temperature range Tm >T >To . In this temperature range the system temporarily attains the -structure with life time larger than the inverse of vibration frequency. For temperatures lower than To (), the -phase is more stable than the bcc phase.

towards the value corresponding to ideal -structure as shown by dotted line in Figure 6.19. As temperature reaches below Tm , a true minimum in free energy is attained and the -structure becomes stable with respect to the bcc structure. The minima correspond to the metastable states in the temperature range T To (). The strongly anharmonic behaviour of Zr with respect to the displacement ordering with the wave vector 2/3 ∗ has been demonstrated by first principles calculations of Ho and Harmon (1990). This point is discussed in the following section. The same conclusion can be arrived at from the fact that the K

504


wave, acting alone, can produce a non-vanishing third-order term, as shown by the summation rule: K + K + K = 222, a reciprocal lattice vector. 6.6.2 Formation of the plate-shaped induced by shock pressure in -alloys The lattice collapse mechanism of the → transformation is pure shuffle transformation and does not envisage any macroscopic shape strain. When such a mechanism is operative, the product and the parent phases can maintain complete coherency along the entire interface. For such a transformation the strain energy minimization by a proper choice of the product morphology is not important. However, when a macroscopic shape strain is superimposed on the → transformation, choice of the interface is restricted by the criteria of minimization of the interfacial energy and the strain energy associated with the transformation. For example, in the martensitic transformation, the strain energy contribution dominates and the invariant plane strain (IPS) condition is satisfied, the surface energy term dictates the selection of habit planes. In several diffusional transformations also the invariant line (IL) vector, lying along the habit plane, which defines the primary growth direction, is essentially due to the minimization of the strain energy associated with the transformation. In the similar fashion, if there is a volume change involved during the → transformation and if the external stress involved has a shear component, the choice of the interface is not completely random. In fact, it has been shown under these circumstances the product -phase has plate-shape morphology and habit plane between the - and the -phases can be predicted using phenomenological theory of martensite (Dey et al. 2004). Using the phenomenological theory of martensitic transformation, Dey et al. (2004) could derive the Bain strain and also the macroscopic shape strain matrix. The predicted habit plane was very close to the experimentally determined (121) habit plane (Figure 6.20(a)–(d)). The observation of {112} planes as the habit planes of the -phase is suggestive of the fact that the → transformation involves a shearing of the -lattice along a {112} plane. A bcc structure can be described by a sixlayer packing sequence of {112} planes, as shown in Figure 6.21. Hatt and Roberts (1960) have examined the possibility of generating the -structure by gliding an {112} plane and have shown that a suitable sequence of glides can indeed produce the → transformation. Figure 6.21 shows that a macroscopic shear on an {112} plane along an direction, superimposed with atomic shuffles, can produce the -structure. Since the macroscopic deformation is a simple shear, the invariant plane, which is the contact plane between the two phases, is the shear plane itself. In this case, the lattice invariant shear operates for the macroscopic strain to satisfy the invariant plane strain condition. It is possible to define a homogeneous shear on the (112) plane with a magnitude of cot /2 = 0767, where


505

Figure 6.20. (a) SAD pattern from a Zr–20% Nb alloy showing the presence of the -phase. The occurrence of diffuse intensity indicates the tendency for the -phase transformation, (b) plateshaped morphology of the -phase, (c) interface between a plate of the -phase and the -matrix is lying along a direction and (d) composite SAD pattern showing the presence of a single -variant.

¯ and the [110] directions. The [1¯ 10] ¯ is defined as the angle between the [1¯ 12] direction (2 ) remains undistorted as the result of application of this shear, which ¯ direction is perpendicular to the plane of is applied in such a way that the [1¯ 12] shear. Using the well-known geometrical relation of homogeneous deformation, which is very commonly used in the case of deformation twinning, the magnitude of shear can be determined. The homogeneous deformation comprising the shear part alone produces a lattice, which is indistinguishable from the -parent lattice. When atomic shuffles similar to those shown in Figure 6.21 are superimposed on the homogeneous deformation, the -structure is generated. The lattice

(111)β⎥⎪(0001)ω

0 1 2 3 4 5 6

Atomic plane no.

ω

[110]

Displacement

[111] [111]

β

[001]

Phase Transformations: Titanium and Zirconium Alloys [112]

506

[112]

β

[111]

Figure 6.21. Planar stacking of the {112} planes of the bcc lattice (shown by open circles). If the shuffling of the atoms is superimposed during the shearing of the {112} planes, the -structure (shown by the filled circles) is produced. The orientation relationship between the two structures remain same as shown in Figure 6.7. The two lattices have also been shown. In order to bring out the equivalence between the shear model and the wave model, the displacement wave has also been shown in the inset.

correspondence between the - and -lattice remains the same as that encountered when a longitudinal displacement wave is introduced in the -lattice. This equivalence, generated by (i) a combination of shear and shuffle and (ii) a longitudinal displacement wave, can be appreciated from Figure 6.21. Since any specific {112} plane contains only one direction, a single orientational variant of the -structure can be produced within a single {112} plate. In short, the deformation produced by a shock wave in the bcc lattice develops a tendency for the shear of the {112} planes where the formation of the -structure occurs when the shuffle process is superimposed on the shear deformation. The axis of the shear direction decides the choice of the variant for the direction contained in the specific {112} plane and, therefore, plate-shaped single variants of the -phase emerge in the shock deformed -phase. 6.6.3 Calculated total energy as a function of displacement It was believed earlier that the -phase occurred exclusively in Ti, Zr and Hf alloys. However, more recently -structures have been reported in other systems as well (de Fontaine, 1988). Nevertheless, these Group 4 metals are the only pure


507

metals in which the -phase is observed. This suggests that the stability of the -phase is directly related to the electronic structure of these Group 4 transition metals, all three of which exhibit an allotropic bcc → hcp transition. The fact that the collapsed plane in the -structure has the same two-dimensional structure as the basal plane of graphite (Group 14) is also indicative of the inherent stability of the collapsed plane configuration for these systems with some similarity in the electronic structure. Ho et al. (1984) have evaluated the total energy as a function of the displacement corresponding to the longitudinal 2/3 phonon for Mo, Nb and bcc Zr on the basis of first principles calculations. The basic procedure for using total energy calculations to study a particular phonon mode includes first principles band structure calculations taking as inputs the atomic number, the crystal structure, the lattice parameter and the atomic displacement corresponding to a particular “frozen phonon”. This involves the assumption of the adiabatic condition which implies that the motion of ions is so much slower than the motion of electrons that at each instant of a given ionic displacement the electrons are in the ground state defined by the instantaneous ionic positions. The large difference between the electronic and the ionic masses makes this assumption quite valid. To study a particular phonon the ionic positions are fixed at various displacements from their equilibrium positions in the direction corresponding to the phonon eigenvector (polarization). A fully self-consistent band structure calculation has been performed by Ho et al. (1984) for each frozen-in condition and the total potential energy has been determined, combining the electronic energy and the ion–ion coulomb energy. The total energy, expressed in terms of the displacement, has been obtained not only for small displacements where harmonic contributions predominate but also for large displacements which correspond to structural transitions. Experimentally determined phonon dispersion curves (Figure 6.22(a)) for the longitudinal branch for three bcc metals, Mo, Nb and Zr (high temperature bcc phase) have been compared (Stassis et al. 1978). Though these elements are neighbours in the periodic table, their phonon frequencies are remarkably different. The pronounced dip seen near 2/3 for Zr is close to that required for inducing the → transition. Frozen phonon calculations for this mode have shown that the total energy versus displacement curves (Figure 6.22(b)) for Mo and Nb are close to quadratic, implying an essentially harmonic behaviour. The curve for Zr is strongly anharmonic and the minimum in the energy occurs when two of the (111) planes collapse together to form the -phase. It is seen that the ground state energy of bcc Zr rises with displacement on the positive- side and after reaching a maximum drops down to a lower minimum at a displacement of 0.5 corresponding to the ideal -structure, while on the negative displacement (anti-omega structure) side the energy rises very sharply. The results of the first

508

Phase Transformations: Titanium and Zirconium Alloys 8 L [111] 7

Nearly harmonic 0.010

Energy (Ry/atom )

ν (THz )

6 5 4 3 2

Mo

Strongly anharmonic

Zr 0

1

kω

Nb

Ideal ω

km bcc

0 0.2

0.4

0.6

ξ (a)

0.8

–0.2

0

0.5

Displacement of (222) planes (b)

Figure 6.22. (a) Phonon dispersion curves for Mo, Nb and bcc Zr showing pronounced dip for Zr at k = k . (b) Calculated total energy as a function of displacement corresponding to the longitudinal (2/3, 2/3, 2/3) phonon.

principles total energy calculations, therefore, are consistent with the Landau plots for the → transition which have been discussed in the preceding section. These calculations also reveal the influence of large atomic displacements on the local charge density. For Zr it has been shown that a transfer of d-like charge from uncollapsed (A type) planes to s-p like charge in collapsed (B and C type) planes occurs which is unlike the calculated behaviour of charges in Nb and Mo. Thus s-p like bonding tends to develop in the collapsed planes of Zr (and other Group 4 metals), mimicking the bonding of atoms in the basal plane of graphite. The calculated total energy, as a function of displacement, shows that the bcc phase is unstable with respect to the → transformation. However, in pure Ti and Zr this transformation cannot be induced without the application of hydrostatic pressure. This is due to the relative stability of the hcp -phase in these systems. The phonon description of the bcc → hcp transition has been given earlier in Chapter 3. The addition of bcc alloying elements (V, Mo, Nb, Cr, Fe, etc.) to Ti and Zr promote the formation of the athermal -phase. This is incidental because these alloying elements actually destabilize the -phase with respect to the -phase but apparently they do so not as strongly as they destabilize the -phase with respect to the -phase. This point is discussed in Section 6.6.4 where -phase stability is rationalized in terms of d-band occupancy by electrons.


509

6.6.4 Incommensurate -structures The lattice collapse mechanism described earlier can account for the first-order nature and the athermal displacement ordering character of the → transition. Landau plots (G versus ) at different temperatures (Figure 6.19) can also provide a description of the increase in the incidence of interplanar collapse with decreasing temperature and the presence of pretransition effects. The perfect -structure, however, cannot explain why the diffuse peak positions are not located exactly at the -reflections and why a fine dispersion of -particles in the -matrix invariably forms in the as-quenched condition. There have been some approaches involving incommensuration of the -structure which attempt to explain the offset of the diffuse peaks and the duplex fine particle ( + ) morphology. Cook (1975) has argued that the offset of the diffuse intensity peaks has its origin in the modulation of the “ideal-” structure by a wave with wave vector, q = Km − K , where Km , as defined earlier, is the wave vector associated with the maximum of diffuse intensity. He has used a formalism, as outlined here, in which the Fourier transform (denoted by K-representation) of the free energy is considered. The displacements, ui , are substituted by a Fourier series: (6.9) ui = UK expiKxi K

in which xi is the distance from the origin to the ith plane of the undistorted planar lattice and UK is the Fourier amplitude given by UK = 1/N

N

ui exp−iKxi

(6.10)

i=1

where N is the total number of planes and K is an allowed wave vector in the first Brillouin zone. The expression for the free energy change, on the substitution of Eq. (6.10) in Eq. (6.7), becomes G = NF2 + F3 + F4 +

(6.11)

where F2 , the purely harmonic portion of the free energy, and the third order anharmonic term F3 are respectively given by F2 =

1 KUK2 2 K 2

(6.12)

F3 =

1 K K UKUK K + K + K g 3! KK K 3

(6.13)

510


In these expressions, 2 (K) and 3 (K) are appropriate K-dependent secondand third-order coefficients, which are essentially the Fourier transforms of the second-order and the third-order interplanar coupling parameters: the term K + K + K g is a delta function which is zero unless the sum (K + K + K ) is equal to a reciprocal lattice vector, g, along an direction. The complex amplitudes U are proportional (in modulus) to the displacements of the atomic planes which are essentially Fourier transforms of the second-order and third-order interplanar coupling parameters: (6.14) 2 K = ii+j expiKxj j

3 K =

j

ii+jj+l expiKxj iKxl

(6.15)

l

For the trivial solution of zero amplitudes for all K, as is the case for the bcc structure, the free energy change G, assumes the minimum of value zero. If the system behaves in an essentially harmonic fashion (i.e. the contributions of F3 , F4 , etc. are negligible), there is no other minimum in G as an increase in the amplitude raises the value of the free energy change parabolically. Body-centred cubic metals such as those belonging to Groups 5 and 6 (e.g. Nb and Mo) exhibit such harmonic behaviour (Figure 6.22). The first principles electronic structure calculations for bcc Zr, discussed in Section 6.6.3, clearly demonstrate a strongly anharmonic behaviour, as revealed by the presence of a deep second minimum away from the zero amplitude first minimum (Figure 6.22). A similar behaviour is also expected in the cases of Ti and Hf. As mentioned earlier, the harmonic contribution (arising out of the second-order force constant, 2 , to the free energy change) is minimized when the wave vector is Km while the anharmonic contribution (arising from the thirdorder term) can further reduce the free energy change through interactions between displacement waves of different wave vectors lying in between Km and K . The fluctuation thus created results from a competition between the second- and the third-order force constants. Cook (1975) has demonstrated this fluctuation mechanism by considering the growth of athermal displacement modulation represented by a sinusoidal standing wave ui = A sinK + qxi = A cos qxi sin K xi + A sin qxi cos K xi

(6.16)

with the amplitude A being related to the Landau order parameter, , by the relation A = a /6

(6.17)


511

Longitudinal displacements

When qxi is small, the first term gives the ideal -structure. As xi increases to a value of /q, the anti- structure develops. The presence of a spectrum of wavelengths from K to Km (= K + q) will generate wave packets of - and anti- structures as shown in Figure 6.23(a). The anti- structure, however, is

+ 0 –

5 1

2

3

6

7

8 9

4

10

Distance in (111) interplanar units (a) ω

–ω

ω

2π/Δk (b) G

G

G

T < To(ω ) Km

Kω

K = Km

K = Kω K = Km

K = Kω A

(β + ω ) Dual structure

A∗ (c)

Aω A

T H B where the subscripts B and H, respectively, stand for the B82 and the D019 structures (Tewari et al. 1999). This observed orientation relation shows that the lattice correspondence between the B82 and the D019 structures compares reasonably well with that between the and the -phases observed in the → transition induced by the application of high pressures, as discussed in Section 6.3.3. The B82 /D019 structural relationship suggests that the D019 structure can be created by the introduction of Zr atoms ¯ plane of the latter from which the (0001) plane in the B82 structure. An (1120) of the former emerges is shown in Figure 6.34. The positions marked by grey circles indicate the vacancies in the B82 structure which are filled up by Zr atoms in the D019 structure. The transformation sequence, Zr5 Al3 (D88 )→ Zr2 Al(B82 )→ Zr3 Al(D019 ), can, therefore, be considered as a process in which an open structure is gradually being filled by an influx of Zr atoms, progressively changing the composition and finally attaining a close packed configuration. Such a process can occur as long as the chemical potential of Zr in the adjoining -matrix is higher than that in the intermetallic phase, which finally attains the stoichiometry of Zr3 Al. The metastable D019 structure of the Zr3 Al phase subsequently transforms into the equilibrium L12 structure through a congruent transformation. This transformation can be accomplished by a change in the stacking sequence from the hexagonal ABABAB to the cubic ABCABCABC stacking of the close packed atomic layers which individually satisfy the Zr3 Al stoichiometry. The low-stacking fault energy (Holdway and Staton-Bevan 1986) (2 mJ/m2 ) of Zr3 Al, which can be taken as a measure of the free energy difference between its L12 and D019 forms, is consistent with the fact that the metastable D019 phase appears first when Zr3 Al precipitates from a supersaturated hcp Zr–Al solid solution (Mukhopadhyay et al. 1979). The whole sequence of formation of various phases has been schematically presented in Figure 6.35. 6.7.3 Transformation sequences in Ti base alloys The sequences of phase transformations involving ordered -phases have been studied primarily in the ternary Ti–Al–Nb system by Strychor et al. (1988), Bendersky et al. (1990a,b) and Bendersky (1994). The pseudo-binary phase diagram of the Ti3 Al–Nb system shows the composition range over which the -phase undergoes a martensitic transformation following which an ordering process leads to the formation of the ordered 2 -phase (D019 structure), as discussed earlier. In Ti3 Al–Nb alloys rich in


531

aH = 0.604 nm cB = 0.592 nm Distortion = 1.82%

[0001]B//[1120]H

[1100]B//[1100]H

√3 aB = 0.8476 nm √3 aH = 1.043 nm Distortion = 18.73%

CH = 0.493 nm aB = 0.489 nm Distortion = 0.82%

B type atoms A type atoms A type atoms missing in B82

¯ > plane of the B82 structure. A systematic Figure 6.34. Schematic drawing of a prismatic 1193 K), DRX became the dominant deformation mode at lower strain rates. Increasing the temperature of deformation enhanced the process of DRX. Both polygonization and DRX were also observed in the -phase in which subgrains were formed. DRX and polygonization was recorded on deformation in both the initial microstructures but the DRX process was dominant in the case of equiaxed morphology. These alloys also exhibited superplasticity with equiaxed morphology on deformation at lower strain rates (≈0.0001 s−1 ) and at temperatures in the range 1198–1223 K. Sastry et al. (1980) carried out detailed TEM investigation of hot deformed Ti–6Al–4V material at a strain rate of 0.05 s−1 and at temperatures above 850 C and reported that both DRX and DRV occurred as evidenced by a hexagonal network of dislocations in the -phase, formation of small equiaxed -grains and absence of shearing in -phase. Recently, Seshacharyulu et al. (2000, 2002) studied the effect of initial microstructures (equiaxed and -transformed) and impurity content on the hot working behaviour of Ti–6Al–4V by constructing processing maps over a wide range of strain rate and temperature. The processing map for the equiaxed morphology exhibited two domains, viz., a domain of fine grain superplasticity in the ( + ) region and another domain of DRX in the -phase field. The ratecontrolling accommodation process during superplastic deformation was identified to be cross slip in -phase which was located at the triple junction of -grains. The activation energy for the process of DRX in the -phase was close to that for self-diffusion in . During deformation in the ( + ) phase field , the -plates of the transformed -structure assumed equiaxed morphology by the process of globularization. The primary -grain size varied linearly with the Z parameter (Zener Hollomon parameter Z = $˙ exp [Q/RT], where $˙ is the strain rate, T is the temperature in K and Q is the apparent activation energy) in a manner similar to that observed during DRX. The deformation in the -phase field resulted in DRX with characteristics similar to that observed in the initial equiaxed microstructure. Chen and Coyne (1976) carried out isothermal forging experiments on Ti–6Al–4V and identified DRX as the rate-controlling deformation process on the basis of the apparent activation energy obtained by kinetic analysis. Weiss et al. (1986)

Diffusional Transformations

689

carried out detailed TEM investigation of the hot deformed microstructure of a Ti–6Al–4V alloy with Widmanstatten morphology and reported that -lamellae break up during deformation either heterogeneously by forming intense shear bands or homogeneously by formation of a dislocation substructure. Depending on the extent of localized shear strain, the -lamellae may sever completely or partially. These sheared regions, during the course of deformation, will either form / grain boundaries or will get converted to a structure in which -regions are separated by -phase by its complete severance by percolation of -phase. During the homogeneous deformation, however, dislocations rearrange to form recovered substructure leading to the formation of subgrains which divide the lamellae into smaller units. The -phase then penetrates along the / sub-boundaries. 7.8.2.3 -alloys Although the -alloys are cold workable and can be recrystallized at temperatures as low as 1073 K to obtain fine grains, the initial bulk working of the ingot is done at temperatures above their -transus followed by secondary working either above or below the transus (Weiss and Semiatin 1998). Numerous hot working studies have been conducted on -alloys both in the single phase as well as two-phase ( + ) regime. The various aspects of thermo-mechanical processing of -Ti alloys have been reviewed by Weiss and Semiatin (1998). These alloys were found to exhibit discontinuous yielding followed by almost a steady-state flow behaviour, formation of serrated -grain boundaries on deformation and the presence of subboundaries ranging from medium- to high-angle boundaries during deformation in the -phase. The deformation in the -field appeared to be controlled by dynamic recovery with an apparent activation energy which matched with that of self-diffusion. Similar conclusion was drawn by Robertson and McShane (1998) who examined the effect of two initial microstructures, viz., -transformed and equiaxed ( + ), on the deformation behaviour of Ti–10V–2Fe–3Al alloy at lower strain rates. Balasubrahmanyam and Prasad (2001, 2002) have characterized the hot deformation behaviour of two -alloys, Ti–10V–2Fe–3Al and Ti–10V–4.5Fe– 1.5Al over a wider range of strain rates and temperatures using processing maps. They found that the processing of -alloy (Ti–10V–4.5Fe–1.5Al) showed a single domain in the temperature range 1023–1173 K and strain rate range 0.01–0.1 s−1 . On the basis of the microstructural features, the variation of grain size of the deformed grain structure (predominantly equiaxed) with temperature and tensile ductility variations, it was concluded that the prevailing deformation mechanism in the domain was DRX. The processing maps of the metastable -alloy (Ti–10V– 2Fe–3Al) on the other hand showed two domains during deformation in the similar temperature and strain rate ranges. In the temperature range 923–1023 K and at strain rates lower than 0.1 s−1 , the material deformed superplastically exhibiting

690


large ductility. The accommodation process was identified to be dynamic recovery from the nature of stress-strain diagram, which was of steady state type, and from the apparent activation energy value, which matched with the activation energy for self-diffusion in -Ti. At temperatures higher than 1073 K and strain rates lower than ∼0.1 s−1 , the alloy exhibited large grain superplasticity (LGSP) with a deformed microstructure containing stable subgrain structure within large -grains. LGSP was also reported to occur during hot deformation of -Ti alloys by Griffith and Hammond (1972). 7.8.2.4 Ti-aluminides Titanium aluminides offer a combination of properties such as low density, high strength to weight ratio, high modulus and good elevated temperature strength properties which make them excellent candidate material for gas turbine engine and airframe structure in advanced aircrafts. The aim during primary ingot breaking of these materials is to prevent catastrophic fracture during shape change as well as to provide suitable homogeneous microstructure to enhance hot workability during secondary operations (Semiatin et al. 1992, 1998). One of the main limitations of these materials is their limited workability during conventional metal working processes such as forging, rolling and extrusion because of the generation of various internal defects. The importance of primary hot working can be emphasized by taking a typical example of a near--TiAl alloy. This variety of aluminides, with aluminium content more than 46%, does not have single phase -field as in the case of conventional titanium alloys. This coupled with significant coring of the dendritic arms that result from the peritectic solidification and presence of ordered phases even at very high temperatures (> 1573 K) make primary ingot breakdown a technological challenge. The hot deformation mechanisms of these materials during secondary processing have been found to be a strong function of the starting microstructure, deformation temperature and strain rate in a manner similar to that observed with conventional Ti alloys. The major issues of thermo-mechanical processing of Ti-aluminides have been reviewed by Semiatin et al. (1998). A considerable amount of effort has been directed to identify the suitable hot working conditions and the associated deformation mechanisms to obtain defect-free products for a variety of single phase 2 , single phase , twophase (2 + ) and near- alloys by developing workability maps as well as by microstructural observation of hot deformed materials. Semiatin et al. (1998) and Huang et al. (1995, 1998) studied the deformation behaviour of cast Ti–24Al–11Nb over a range of temperature and strain rate and identified three different regimes of deformation: a regime of warm working where deformation led to distortion of -laths, wedge cracking and cavity formation, a region of globularization of -laths in the two-phase (2 +) and a domain of hot working in the single phase .


691

Sagar et al. (1994, 1996) studied the hot deformation behaviour of a similar alloy in two different initial microstructural states employing dynamic materials model. They identified a domain of DRX of 2 -phase in both the starting microstructures. In addition, they have also observed a second domain of DRX and a domain of superplastic deformation of -phase for different combinations of strain rate and temperature. The flow behaviour and microstructural development during forging of super 2 (Ti-25Al-10Nb-3V-1Mo) have been studied by Huang and co-workers (1995, 1998). It was found that while both DRV and deformation heating were involved in microstructural evolution during hot deformation of (2 + ) alloys at rates lower than 10 s−1 , the dominant deformation mechanism was DRX at rates higher than 10 s−1 . However, both DRX and DRV coupled with certain amount of grain boundary sliding were responsible for flow softening during deformation in the -phase field at both high and low rates of deformation. In view of very restricted workability of large lamellar microstructure of aluminides, several TMP methods are being increasingly employed to break down the initial microstructure to smaller units (Martin 1998, Sun et al. 2002). For microstructural refinement of orthorhombic Ti–22Al–27Nb, Martin (1998) used near-isothermal processes involving rolling at temperatures just below the -transus followed by a thermal treatment to obtain a fine grain metastable -phase. The lamellar microstructure of near- Ti–Al, which does not have an overlying -phase, is broken down by extrusion in the 2 + phase field by dynamic recrystallization. Subsequently further microstructural refinement is achieved by multiple isothermal forgings. Recently the technique of hot-die incremental deformation has been used with intermittent recovery to breakdown the large 2 + lamellae into fine 2 + microstructure (Pan et al. 2001). Although the hot deformation mechanisms that are involved in conventional Ti alloys are also present in the titanium aluminide alloys, the window of hot working of Ti-aluminides in the as-cast condition is very restricted because of the ease of intergranular fracture, cavitation and flow localization even at higher temperatures. An initial thermo-mechanical treatment to break up the lamellae into fine microstructure is therefore essential for good workability, with the added capability of achieving even superplasticity. 7.8.3 Hot working of Zr alloys The number of commercial alloys of Zr is considerably smaller than that of Ti. Zr-based alloys are chosen primarily for the manufacture of some critical structural components in nuclear reactors. As has been indicated earlier, these alloys are fabricated by processing routes, involving a melting stage, hot working in two stages and cold working with or without annealing. It has been observed by Chedale et al. (1972) that the scale of the final microstructure and texture are determined primarily in the second hot working stage which is extrusion. It has

692


been shown that the extrusion speed and temperature are two important parameters which have significant effect on the development of texture, microstructure and mechanical properties of the finished products (Chedale et al. 1972, Konishi et al. 1972). It is therefore essential to optimize them in order to arrive at the desired final properties. 7.8.3.1 and near--Zr alloys Jonas and co-workers (Lutton and Jonas 1972, Abson and Jonas 1973) studied the high-temperature deformation behaviour of -Zr and -Zr-tin alloys in compression in detail and examined the microstructural development by TEM. They concluded dynamic recovery to be occurring in the temperature range 898–1098 K and strain rate range 10−4 to 3 × 10−3 s−1 on the basis of the observed shape of stress–strain diagram, kinetic analysis performed and detail TEM investigation of deformed microstructure. Although the deformation was carried out at or above the recrystallization temperature (static), no recrystallized grain has been observed in the deformed material. The grain shape in general was elongated and was found to contain sub-boundaries. The mean size and perfection of these boundaries decreased with decreasing temperature, increasing strain rate and increasing tin content. On the other hand, Ostberg and Attermo (1962) carried out forging of Zircaloy-2 (a near--alloy) at 1073 K and at a strain rate of 0.4 s−1 and observed that the original Widmanstatten -plates were broken up into new grains which were equiaxed. They concluded that DRX was involved in the modification of the microstructure. It may be noted that the deformation of Zircaloy-2 at 1073 K is in the -phase and deformation at temperatures higher than 1083 K and up to -transus will be essentially in two-phase ( + ). Garde et al. (1978) studied hot tensile deformation behaviour of zircaloy-2 in the two-phase ( + ) field at various strain rates. The microstructure evolved during deformation at temperatures around 1123 K and lower strain rates (< 10−3 s−1 ) was characterized by equiaxed -grains separated by -phase. The phenomenon of grain boundary sliding has been found to be responsible for the occurrence of equiaxed grain structure. When the deformation rates are increased (≈0.1 s−1 and at the same temperature), elongated -grain structure separated by a thin film of is usually seen. Ostberg and Attermo (1962) have observed similar elongated morphology of separated by -films on forging at 1173 K and 0.4 s−1 (estimated strain rate). A few -plates have been found to change from single crystals to fragmented structure of new grains/subgrains as evidenced by difference in contrast under polarized light. The formation of new grains from the original -plates into smaller units is suggestive of the occurrence of dynamic recrystallization process in the -phase of the two-phase ( + ) mixture. Transgranular deformation mechanisms involving dislocations are important at these rates of deformation to produce the elongated


693

morphology. These observations clearly indicate the effect of strain rates on the mode of deformation and evolution of microstructure. As regards the effect of initial microstructure, an equiaxed grain structure has been found to exhibit better hot ductility than a Widmanstatten microstructure. In ( + ) material of zircaloy-4, the nickel-free variety of zircaloy-2, a hot working activation energy of 155 kJ/mol was reported in the temperature range 840–970 C and at strain rates greater than 3 × 10−3 s−1 (Rosinger et al. 1979). This value of activation energy was similar to that obtained in -phase. Zr and zircaloy-2, when deformed in the -phase field at low rates ≈ 10−3 s−1 , exhibit superplasticity and high ductility (Garde et al. 1978, Bocek et al. 1976). Chakravartty et al. (1991, 1992a,b,c) studied the hot deformation characteristics of commercial pure Zr (containing approximately 1000 ppm of oxygen) with a -quenched starting microstructure in both - and -phase fields using processing maps generated by carrying out compression testing in the strain rate range of 10−3 –102 s−1 and over the temperature range of 650–1050 C. The processing map revealed (Figure 7.78) two safe domains and a regime of instability: (a) a domain of DRX in the temperature range of 730–850 C and strain rate range of 10−2 –0.1 s−1 with its peak efficiency of 40% at 800 C and 0.1 s−1 which was considered as optimum hot working parameters, (b) a domain of dynamic recovery at temperatures lower than 700 C and strain rates lower than 10−2 s−1 and (c) a regime of flow instability at strain rates higher than 1 s−1 and temperatures above 670 C. The characteristics of dynamic recrystallization were found to be similar to those of static recrystallization regarding the sigmoidal variation of grain size (or hardness) with temperature, although the DRX temperature was much higher than the static recrystallization temperature 873 K. Within the domain of DRX, the grain size increased with increase of temperature and decrease of strain rate (Chakravartty 1992, Chakravartty et al. 1992b). The manifestation of instability was in the form of localized shear band (Figure 7.80). During hot working, the regime of instability should be avoided. The processing map of Zr in -phase showed only a single domain in the temperature range 925–1050 C and at strain rates lower than 0.1 s−1 with the following features: (a) the efficiency of power dissipation is high (≈ 60%) with a corresponding high strain rate sensitivity of flow stress (0.45), (b) the stress strain curves show steady-state behaviour with no flow softening and (c) flow stress values are generally low. On the basis of these informations, the domain has been interpreted to represent LGSP in -Zr (Chakravartty et al. 1992a,b,c). Hot working characteristics of zircaloy-2, with -transformed initial microstructure, have been investigated by Chakravartty et al. (1992b) using processing maps in the temperature range of 650–950 C. In this range, zircaloy-2 exists either as predominantly -phase or as a mixture of and . The processing maps

694


125 μm

Figure 7.80. The optical micrograph of Zirconium samples deformed at 700 C and 100 s−1 showing adiabatic shear band across which a variation of microstructure can be seen. The instability parameter assumes negative value under the conditions of deformation.

(Figure 7.81) exhibited three distinct safe domains in the temperature range studied with a discontinuity at around 850 C. The domains occur at 800 C and 0.1 s−1 , 650 C and 10−3 s−1 and 950 C and 10−3 s−1 which have been characterized to be the domain of DRX, the domain of DRV and the domain of superplasticity (marked SPD), respectively. It has been reported that the initial -transformed microstructure is converted to equiaxed grains of during the process of DRX. Besides the domains of stable flow, at strain rates higher than 1 s−1 and temperature of 700–850 C, zircaloy-2 exhibits microstructural instability in the form of adiabatic shear band. A comparison of the processing maps of zircaloy-2 and commercially pure Zr in -quenched condition reveals that the maps are strikingly similar. The DRX domain of both the maps extend over similar strain rate and temperature ranges, although the peak efficiency (38%) for DRX for zircaloy is slightly lower than that of Zr (42%). The addition of tin to Zr as in zircaloy-2 lowers the stacking fault energy of Zr (240 mJ/m2 ) significantly (Sastry et al. 1974) while the diffusion coefficient is not greatly altered (16 × 10−16 m2 /s in -Zr while 3 × 10−17 m2 /s in Zr–1.03Sn alloy) (Chakravartty 1992, Chakravartty et al. 1992b). A simple model was used to explain the DRX characteristics of both Zr and zircaloy-2 (Chakravartty 1992). Of the two competitive processes involved in DRX, namely formation of interfaces (the nucleation step which involves dislocation generation and their simultaneous recovery) and migration of a large-angle boundary, it was found that the process of formation of interface is the rate-controlling step in both these materials. The thermal recovery by diffusion-controlled climb plays

Diffusional Transformations Zircaloy- 2 (β-quenched)

695 Strain = 0.4

2 2 1

Log strain rate

18

0

7

11 21

25

30

11 38 DRX

–1

38 35

44

–2 30

25 30 DRV 35 –3 650 710

SPD 49 56 64

25 770

830

890

950

Temperature (°C)

Figure 7.81. Processing map of -quenched zircaloy-2 for a strain of 0.4 showing various domains: (a) domain of dynamic recrystallization (marked DRX), (b) domain of dynamic recovery (marked DRV) and (c) domain of superplasticity (marked SPD). The number against each contour indicates percentage efficiency.

an important role in both these materials (Chakravartty 1992, Chakravartty et al. 1991). The characteristics of dynamic recovery and flow instability in both these materials were also similar (Chakravartty et al. 1991). The variation of grain size with temperature and strain rate, observed within DRX domain of zircaloy2, would permit microstructural control. In the domain of superplasticity, the efficiency value was high and the overall deformed grain structure was equiaxed. In order to validate the processing maps of zircaloy-2, extrusion trials have been carried out on full size billets and ingots used for making various zircaloy-2 products for nuclear reactor use. Two different temperatures (720 and 800 C) and various extrusion ratios and ram speeds were chosen (Chakravartty et al. 1991). The experimental extrusion conditions were located on both the processing map (Figure 7.82) and instability map (Figure 7.83). Most of the extrusions carried out at 800 C are within the DRX domain and the extrusions carried out at 720 C fall in the instability regime. The extrusion carried out at 800 C showed dynamically recrystallized grains while the microstructure of extruded material depicted signs of flow localization (Chakravartty et al. 1992a,b,c).

696

Phase Transformations: Titanium and Zirconium Alloys Zircaloy- 2 (β-quenched) 2

Extrusion at 720°C Extrusion at 800°C 5

9

Log strain rate

1

13 13

0 39

–1

20 24

DRX 28

–2

32 28

32

35 39 DRV –3 650 680

24 730

20

770

810

850

Temperature (°C)

Figure 7.82. Extrusion conditions superimposed on processing map of -quenched zircaloy-2 for a strain of 0.4. The extrusions carried out at 800 C are within the DRX domain while those carried out at 720 C are outside the DRX domain.

7.8.3.2 + alloys Jonas and co-workers (Jonas et al. 1979, Choubey and Jonas 1981, Rodriguez et al. 1985) studied flow properties of Zr–2.5Nb alloys by constant true strain rate compression testing in the strain rate range 10−4 to 1 s−1 and at temperatures from 750 to 1000 C. The alloys showed considerable flow softening in the ( + ) phase region and steady-state flow behaviour in single phase -region. The strain rate sensitivity was found to increase from 0.18 during deformation in the (+) phase field (800–875 C) to 0.22 in the -phase field (900–1000 C). The mechanism involved during hot deformation, however, has not been identified. Zr–2.5Nb alloys have been found to exhibit superplasticity in the temperature range 720–850 C and at strain rates in the range 10−2 –10−4 s−1 (Nutall 1976, Holm et al. 1977, Rosinger et al. 1979). Superplasticity has also been reported in cold worked and stress relieved Zr–2.5Nb pressure tube material during tensile deformation in the temperature range of 650–800 C at strain rates lower than 10−3 s−1 (Singh et al. 1993). The development of microstructure during high-temperature tensile deformation of Zr–2.5Nb pressure tube (hot extruded and cold worked) has been studied by Perovic et al. (1985) using TEM. At deformation temperatures lower than 600 C, the starting lamellar morphology is retained but the original dislocation structure is destroyed and is replaced by sharp sub-boundaries which fragment

Diffusional Transformations Zircaloy- 2 (β-quenched)

697 Strain = 0.4

2

1 –25

–63 –38

–88

–13

Log strain rate

0.00 0

–1

SAFE

–13 0.00

Extrusion at 720°C Extrusion at 800°C

–2

–3 650

680

730

770

810

850

Temperature (°C)

Figure 7.83. Extrusion conditions superimposed on instability map of -quenched Zircalloy-2 for a strain of 0.4. The extrusion carried out at 800 C are in the stable flow regime while those carried out at 720 C are within the instability regime.

the elongated -plates. When deformation is carried out at temperatures lower than -transus, the initial lamellar structure changes over to an approximately equiaxed structure of - and -grains. In this temperature region (>700 C and < -transus) during tensile deformation of Zr–2.5Nb in equiaxed ( + ) microstructure, Nutall (1976) observed that initial equiaxed morphology is maintained even after large deformation (>500%) and suggested superplasticity to be the deformation mechanism. The development of microstructure during hot extrusion of Zr–2.5Nb has been studied by Chedale et al. (1972) and Perovic et al. (1985). The temperature and extrusion ratio have been found to be the two major factors that influence the microstructural evolution in the billets. The extrusion of the -quenched billet at 780 C results in a microstructure in which ( + ) grain structure has been aligned by working. The -constituent is present in the form of lenticular plates which are joined together by sub-boundaries. The hot deformation behaviour of two-phase ( + ) alloys, Zr–2.5Nb and Zr– 2.5Nb–0.5Cu in the temperature range 650–1050 C and at strain rates in the range 0.001–100 s−1 have been studied by Chakravartty et al. (1995, 1996) using processing maps generated by carrying out compression testing. Two different initial

698


microstructures of Zr–2.5Nb alloy, namely equiaxed ( + ) and -transformed (Chakravartty et al. 1992a,b,c), were studied. In both these materials, a domain of DRX has been identified in the processing maps by noting variation of the efficiency values with temperature and strain rate, nature of flow behaviour and by carrying out detailed metallographic investigations using both optical and transmission electron microscopy (Figure 7.84). While a steady-state flow behaviour has been observed in the equiaxed ( + ) material, with a peak efficiency of 45% at 850 C and 0.001 s−1 , the -transformed material exhibits stress–strain curves with continuous flow softening and higher peak efficiency (>50%) under similar deformation conditions. Deformation of equiaxed ( + ) structure in the DRX domain results in the development of equiaxed -grains distributed in a matrix of . A majority of these grains have been found to be further subdivided into subgrains separated by twist or tilt boundaries. In the DRX domain of -transformed material, speroidization of deformed microstructure occurred as a result of the shearing of -platelets followed by globularization (Figure 7.85). This observation is similar to that observed by Weiss and co-workers (1986) during two-phase deformation of ( + ) titanium alloys. A simple model has been used to analyse the characteristics of DRX by considering the rates of nucleation and growth of recrystallized grains (Prasad and Ravichandran 1991). Calculations of these two rates show that they are nearly equal and that the nucleation of DRX is essentially controlled by mechanical recovery involving cross slip of dislocations (Chakravartty et al. 1996). Tensile tests carried out over a range of temperatures and at a strain rate of 0.001 s−1 showed a peak in ductility within the DRX domain. The optimum hot working temperatures for equiaxed ( + ) and -transformed preform microstructures are 850 and 750 C, respectively, while the optimum strain rate is 0.001 s−1 for both. It may be noted that the DRX temperature 850 C, of Zr–2.5Nb, is higher than in -Zr (800 C) and this increase is expected in view of the back stress generated by the long-range elastic interactions of the solute. Further, in comparison with the processing map for -Zr, the map for Zr–2.5Nb showed that the DRX domain has shifted to lower strain rates by two orders of magnitudes (from 0.1 s−1 in -Zr to 0.001 s−1 in Zr–2.5Nb). The lowering in DRX strain rate has been attributed to an increase in the probability of recovery or a decrease in link length of dislocation or both. In view of likely increase of stacking fault energy and solid solution strengthening caused by niobium, it is possible that both these contributions are responsible for the lowering of the strain rate for DRX (Chakravartty et al. 1996). Figure 7.86 shows a TEM photograph of the dynamically recrystallized Zr–2.5Nb which exhibits the extensive formation of twist boundaries due to enhanced cross slip of screw dislocations and its role in forming the interfaces. Kapoor and Chakravartty (2002) studied


699

Zr–2.5Nb (E)

Strain = 0.4

2 3 8 1

14

Log strain rate

19

24

0

29

–1

35 –2

40 45

–3 650

710

770

830

890

950

Temperature (°C)

(a) Zr–2.5Nb (Q)

Strain = 0.4

2 7 10 17 1

21

Log strain rate

24 0 28 31 –1 35 38 42

–2

45 49 52 –3 650

710

770

830

890

950

Temperature (°C)

(b)

Figure 7.84. Processing maps for (a) + equiaxed starting microstructure and (b) -transformed starting microstructure of Zr–2.5Nb for a strain of 0.4. That the maps show a single domain in the ( + ) phase filled for a strain rate lower than 5 × 10−1 s−1 . The number against each contour indicates percentage efficiency.

700


α

β

α 1.2 μm

Figure 7.85. TEM micrograph of -transformed microstructure of Zr–2.5Nb deformed at 800 C and 0.001 s−1 to a strain of 0.3, showing initial stages of dynamic recrystallization involving fragmentation and shearing of -plates followed by globularization.

α

α

β

0.4 μm

Figure 7.86. TEM micrograph of equiaxed ( + ) microstructure of Zr–2.5Nb on deformation at 800 C and 0.001 s−1 . The dynamically recrystallized -grains are found to contain twist boundaries in the form of hexagonal network of screw dislocations. Enhanced cross slip of screw dislocations results in the formation of interface required for dynamic recrystallization.

deformation behaviour in the -phase field of Zr–2.5Nb alloy and reported LGSP at temperatures greater than 950 C and at strain rates lower than 0.01 s−1 . The instability map generated revealed that Zr–2.5Nb alloy undergoes microstructural instability at temperatures lower than 700 C and strain rates higher than 1 s−1 and the manifestation of the instability has been in the form of flow


701

localization in both the initial microstructures and the severity of flow localization has been found to increase with rate of deformation (Chakravartty et al. 1996)). In Zr–2.5Nb–0.5Cu, copper lowers the to transition temperature and partitions entirely to -phase. It is expected therefore that copper modify the deformation characteristics of -phase alone in the alloy. The deformation behaviour has been studied in the temperature range of 650–1050 C encompassing both the ( + )and -phase fields (transformation temperature for this alloy being 870 C). The hot deformation characteristics of this material is similar to that of Zr–2.5Nb in the ( + ) phase deformation in terms of flow behaviour and occurrence of DRX domain. The domain of DRX is seen with its peak efficiency at about 750 C and 0.001 s−1 (Chakravartty et al. 1995). In addition to the DRX domain, another domain was found in the -phase field of this material at 1050 C and 0.001 s−1 which has been characterized to be that of LGSP (Chakravartty et al. 1995). The instability map reveals that microstructure instabilities will occur at temperatures greater than 800 C and strain rates higher than 30 s−1 . 7.8.3.3 -alloys The deformation characteristics of -Zr-Nb alloys (containing 10–20% Nb) have been investigated by Jonas and co-workers (1979) in the temperature range from 725 to 1025 C and with strain rate range 10−5 –10−1 s−1 to assess flow properties of the metastable -phase in the two-phase ( + ) field of Zr–2.5Nb alloy. The flow curves obtained on Zr–Nb alloys exhibited flow softening, and magnitude of this effect decreased as the test temperature is increased. The occurrence of flow softening has been attributed to the disintegration or over-ageing of Nb clusters during straining. 7.8.4 Development of texture during cold working of Zr alloys The overall fabrication schedule which consists of one or two hot working steps followed by a series of cold working steps determine the microstructure of the finished product. The development of crystallographic texture during the process of TMP can again be divided in two parts – the first being produced during hot working operation while the second resulting from the cold working and intermediate annealing steps. The anisotropic crystal structure of -phase makes it quite sensitive to texture development which contributes significantly towards anisotropy of mechanical and other physical properties of the finished product. Although similar deformation characteristics of Ti and Zr result in similar textural development during TMP in their alloys, the texture development in Zr alloys has attracted greater attention owing to the fact that the texture plays a significant role in-service performance of Zr alloys in irradiation environment. A brief account of texture development of zirconium alloys during deformation processing

702


and its importance in mitigating some life-limiting degradation mechanisms in actual components is given here. One of the prime concerns in deformation processing is the development of a favourable crystallographic texture which is usually represented in terms of the distribution frequency of important crystallographic planes (e.g. basal or prismatic planes) with reference to the axial, radial and circumferential directions for tubular products and to the rolling, normal and transverse directions for plate products. As both slip and twinning contribute to plastic deformation of Zr, the deformation texture is strongly dependent on the process variables such as rate of deformation, temperature, extent of deformation and state of stress (Tenckhoff 1988). These variables in turn dictate available slip/twin systems and their respective critical resolved shear strength. The development of texture with increasing degree of cold work is due to the lattice rotations caused by a number of deformation systems. Twinning causes spontaneous lattice rotations through large angles though the resulting strain is small. Consequently at low values of plastic strain, the contribution of twinning is more significant in texture development. After large plastic strains, texture development is mainly controlled by slip deformation which tends to rotate the basal pole by ≈ 20 –40 from the radial normal direction towards the circumferential/transverse direction, provided the major compressive deformation is along the former (Tenckhoff 1988). In the production of zircaloy tubing, tube reduction processes, such as pilger milling involving triaxial stresses and strains, are commonly employed. The reduction in the cross-section (RA) is achieved by reduction in the wall thickness (RW) and the diameter (RD). Depending on the value of the ratio Q (= RW/RD), different crystallographic textures are developed in pilgered tubes. A high wall thickness to diameter reduction (Q > 1) produces a texture with the basal pole aligned parallel to the radial direction while a deformation with Q < 1 leads to a texture with the basal poles aligned parallel to the circumferential direction. Cold rolling of sheets also produces a similar texture with the basal poles being about 20 –40 away from the normal direction. Texture development in different types of metal-forming operations in zirconium alloys is schematically illustrated in Figure 7.87. The crystallographic texture of zirconium alloys influences the physical, mechanical and corrosion properties both out-of-pile and in-pile. Under neutron irradiation, single crystals of -zirconium shrink along the c-axis and dilate on the basal plane with the volume remaining more or less constant. This means a tube with a predominant circumferential basal pole texture will grow along the axial direction under irradiation. In order to minimize such growth, the fraction of basal poles along the axial direction needs to be increased. Similarly the creep strength along the hoop and axial directions of the cladding can be tailored by appropriately


Type of deformation

Body dimension initial–final

703

Schematic (0002) pole figure

Element

AD RD

AD

T

TD

u b

TD RW >1 RD

e RD

r

AD

AD

e d u c

TD

TD RW =1 RD AD

t

RD

AD

i o

TD

TD

n RW 50 5.9 >50

573 >873 823 >1023

where the subscript ‘M’ refers to H dissolved in metals. Sievert’s law for the concentration of dissolved H, HM is

HM = pH2 1/2 exp−G/RT

(8.2)

where G is the standard free energy change for the reaction, pH2 is the partial pressure of H and R and T have their usual meanings. The corresponding enthalpy change, H, has two components: (a) the enthalpy of dissociation of the H molecule (about 52 kcal/mole) and (b) the enthalpy of solution Hs of H atoms in the metal lattice. Depending on the sign of Hs , metals are classified as endothermic and exothermic occluders of hydrogen (Dutton 1976). This classification serves to distinguish the stable phase formed when the solid solubility limit is exceeded. The endothermic occluders are in equilibrium with the H gas, often in the form of internal gas bubbles, and the exothermic occluders are in equilibrium with a precipitated solid hydride phase. Ti and Zr are exothermic H occluders, the enthalpy of solution and the maximum solid solubility limit of H in the (hcp) and in the (bcc) phases being indicated in Table 8.1. 8.2.1 Ti–H and Zr–H phase diagrams The equilibrium phase diagrams of the Ti–H and Zr–H systems are shown in Figures 8.1(a) and (b) in each of which the metastable -hydride phase field is also marked. The two equilibrium hydride phases are the -hydride phase and the -hydride phase. In both the systems the -hydride phase results from an eutectoid decomposition of the -phase, as indicated in the phase reaction = +

(8.3)

The equilibrium -hydride (fcc) phase is stable in the composition range of 56.7– 66.0 at.% H (Zuzek 2000). In this phase H atoms occupy some of the tetrahedral interstices at ( 41 41 41 ) and equivalent positions. Only a very small fraction of H atoms are located in octahedral holes. The volume misfit of -hydride with respect to -Ti and -Zr is about 24% and 17.1%, respectively.

Interstitial Ordering

723

Hydrogen (at.%) 0 10 20

30

40

50

Hydrogen (at.%)

60

0 10 20

1173

40

50

60

1273

1073

1173

β-Ti

973

1073

873

Temperature (K)

Temperature (K)

30

773 673 573

573 K

δ 1.33

0.15

473

2.16

α-Ti

373 273

ε

173

β-Zr

δ

973 873

823 K

773

0.07

1.43

~0.659

ε

α-Zr

673 573 473 373

73

273

–27 0 Ti

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Hydrogen (Wt%) (a)

0 Zr

0.2 0.4

0.6 0.8

1

1.2

1.4 1.6 1.8

2

Hydrogen (Wt%) (b)

Figure 8.1. (a) Ti–H and (b) Zr–H phase diagrams.

The tetragonal -hydride phase has a c/a ratio less than unity and a stoichiometry close to TiH2 and ZrH2 . H atoms occupy the interstitial sites of tetragonal symmetry at ( 41 41 41 ) or equivalent points. There are two such sites per Zr/Ti atom. While the radius of the interstitial site in the -hydride structure is 0.0378 nm that in the -hydride structure is 0.043 nm (in the case of Zr hydrides). The atomic radius of H is about 0.040 nm which matches closely with the sizes of the interstitial sites in both and hydrides. The metastable -hydride phase has a fct structure with a c/a ratio greater than unity and a stoichiometry close to ZrH and TiH. The lattice parameters of this phase depend on the alloy composition and the thermal treatment. Typically, the -hydride phase in a dilute Zr–H alloy has lattice parameter values of a = 04596 nm and c = 04969. H atoms occupy ordered tetrahedral interstitial sites in a fct Zr lattice. The hcp phase of Ti and Zr can take a certain amount of H in solid solution. There are two types of interstitial sites in the hcp structure – tetrahedral and octahedral (Figure 8.2); their radii are 0.0364 and 0.068 nm, respectively, in -Zr. For titanium the sizes are 0.0343 and 0.0619 nm, respectively. Neutron diffraction experiments have shown that H atoms occupy tetrahedral interstitial sites in the phase of Ti and Zr (Narang et al. 1977). There are four such sites of tetrahedral coordination at ( 23 13 61 ), ( 23 13 65 )(00 13 ) and (00 23 ). Darby et al. (1978) in a theoretical study, have calculated the relative energy of H in tetrahedral and octahedral sites in the hcp -phase assuming that the energy terms most strongly dependent on

724

Phase Transformations: Titanium and Zirconium Alloys a√2

a√3

a

2√2

a

Metal atoms Tetahedral interstices

Metal atoms Octahedral interstices (a)

(b)

A

B

C a0

3√

a0 (c)

Figure 8.2. (a) Octahedral, (b) tetrahedral interstitial sites in hexagonal close packed structures and √ (c) a single interstitial layer described in terms of a hexagonal net with transalation of 3ao .

site symmetry are those pertaining to the electronic and protonic electrostatic contributions. Their calculations have shown that the tetrahedral site is preferred over the octahedral site in agreement with experimental observations. There have been several attempts to predict H solubility in the -phase. Using a strain energy-based model, Sinha et al. (1970) have obtained a saturation solubility of 62 at.% H in -Zr at room temperature. However, the experimentally observed maximum solubility limit is about 6 at.% at the eutectoid temperature (823 K) in the Zr–H system. In a thermodynamic study (Northwood and Kosasih 1983), it has been assumed that hydrogen atoms occupy only alternate sites in the -phase and that they oscillate between the two interstitial sites with an estimated vibration frequency of 121 × 1013 Hz. Sinha et al. (1970) have used this concept in modifying their strain energy model but the results have still shown a solubility limit much higher than the experimentally observed value. Considering H as a quantized oscillator, a solubility limit of 0.492 at.% at room temperature has been obtained, which is in good agreement with the observed value (Sinha et al. 1970).


725

8.2.2 Terminal solid solubility The maximum hydrogen concentration, which can be retained in solid soltuion without forming hydride precipitate, is called terminal solid solubility (TSS) (Pan et al. 1996). The amount of hydrogen in excess of solid solubility precipitates as one of the hydride phases depending on the cooling rate and the hydrogen concentration (Northwood and Kosasih 1983). Similar to any solid state phase transformation process, the free energy of hydride precipitation has three components, viz. chemical free energy, interfacial energy and strain energy. The driving force of chemical free energy change associated with the transformation of matrix to hydride is primarily opposed by the strain energy arising from the density difference between the matrix and hydride precipitate. The interfacial energy is negligibly small when hydride precipitates are fine (1.0 ) can coexist with the as well as with -hydride phases. The formation of the -hydride phase is favoured by a fast cooling rate from a temperature where H is taken into the solid solution (typically, above about 625 K for Zr containing about 40 ppm H). Northwood and Kosasih (1983) have suggested a shear mechanism for the → transformation. The temperatures at which -hydride can form are too low for any significant self-diffusion of Zr (or Ti) atoms to occur. Interstitial diffusion of H atoms within the time scale of the transformation is still possible. The overall → transformation process can, therefore, be viewed as a shear transformation of the hcp lattice of -Zr/Ti with an accompanying H redistribution and interstitial ordering. The morphological features (Figure 8.4(a); Dey et al. (1982) Dey and Banerjee (1984)) of -hydride precipitates in the -matrix include plate shape with specific habit plane and internal twinning which suggest an invariant plane strain

200 nm

(a)

Figure 8.4. Micrographs showing -hydride in (a) - and (b) -phases.

300 nm

(b)


729

transformation. The crystallography of the transformation is, therefore, analysed in terms of the phenomenological theory of martensitic crystallography. The -hydride phase can also form in the matrix of -stabilized alloys (Flewitt et al. 1976, Dey et al. 1984). The → transformation also exhibits several features of shear transformations such as low transformation temperatures (at which self-diffusion of Zr or Ti is negligible), a plate morphology and internal twinning of hydride precipitates. In + alloys, H is partitioned between the two phases. Since the solubility of H in the -phase is much more than that in the -phase, precipitation of hydrides in + alloys is encountered more frequently in the -phase and along / boundaries than in the -phase. 8.3.2 Lattice correspondence of -, - and -phases Based on the observed orientation relations between the -hydride and the - and the -phase matrix, Dey et al. (1984) have proposed the following lattice correspondence in respect of these phases (as shown in Figure 8.5). ¯ 1120 ¯ 101 ¯ X1 001 [110]β [1100]α ½[121]γ X2

[111]β [1210]α [110]γ

X1 X3

[001]β [1120]α ½[101]γ

[110]β [0001]α ½[111]γ

Figure 8.5. Lattice correspondences between -, - and -hydrides.

730


¯ 1100 ¯ ¯ X2 110 121 X3 110 0001 111 It may be noted here that while studying the precipitation of -hydride from the -phase, Weatherly (1981) has noted -hydride precipitates of two distinct classes:

¯ habit plane and the orientation (a) Type I -hydride precipitates with 1017 ¯ [12 ¯ 10] ¯ . relationship (111) (0001) ; [110]

¯ (b) Type II -hydride precipitates with 1010 habit plane and the orientation ¯ [1210] ¯ . relationship (001) (0001) ; [110] The lattice correspondences for both type I and type II -plates could be regarded as crystallographically equivalent if one considers the to transition to have two components: the first being the to transition in accordance with the Burgers correspondence and the second being responsible for converting the structure to the -hydride structure. The next two sections have, therefore, been devoted to crystallographic descriptions of the → and the → transformations, respectively. 8.3.3 Crystallography of → transformation In the crystallographic analysis presented here the orthogonal basis vectors ¯ , [110] ¯ [110] , respecx1 , x2 , x3 are taken to be parallel to the directions [001] tively. These vectors are shown in Figure 8.5 which depicts how the (110) plane can be distorted to the (111) plane. The correspondence matrices relating the chosen axial system (xi ) on one hand and the bcc () and fct () axial systems on the other, represented by x RB and x RF , respectively, are given by ⎡

0 B ⎣0 xR = 1¯

⎤ 1¯ 1 1 1⎦ 0 0

⎡

and

1 F ⎣ 0 xR = 1¯

⎤ 1 1 2¯ 1 ⎦ 1 1

(8.11)

The distortion associated with the → transformation can be conceptually divided into two components, S1 and S2 . While S1 is responsible for straining the (110) plane to match the dimensions of the (111) plane and for adjusting the interplanar spacing of the former phase to be equal to that of the latter, the second component, S2 , is necessary for effecting the necessary change in the stacking sequence. The principal distortions, 1 , corresponding to S1 , can be worked out from the lattice correspondence shown in Figure 8.5, to be the following:


[110]β

(110)β

731

B

B

A

A

A

C

B

B

B

A

A

A

[001]β

(0001)α

(111)γ

BCC Distorted by S1

HCP Distorted by S2

FCT

(a)

(b)

(c)

Figure 8.6. Simple shear acting homogeneously on every atomic plane.

√ Along x1 1 = a /2 2a √ Along x2 2 = a /2 3a √ √ Along x3 3 = 2a / 3a The shear, S2 , which changes the AB AB type stacking of the distorted (110) planes (distorted by S1 ) to the ABC ABC type stacking of the (111) planes, is a simple shear acting homogeneously on every atomic plane (Figure 8.6). The shear S2 , being a pure shear, can be expressed as: ⎡ ⎤ 1 0 0 S2 = ⎣ 0 1 03535 ⎦ (8.12) 0 0 1 The total lattice distortion matrix can, therefore, be expressed as S = S1 · S2 where both S1 and S2 are on the basis of the axial system, xi as indicated in Figure 8.5. In order to consider a specific example, the lattice parameters of the -hydride phase and of the -phase in a Zr-20% Nb alloy are substituted for obtaining the lattice strain matrix, S1 , which can be expressed as: ⎡ ⎤ 09734 0 0 S1 = ⎣ 0 11322 0 ⎦ (8.13) 0 0 10856

732


Rigid body rotation Habit plane trace B′

After S1

110

B

Y

Before S1

Undistorted cones

(110)trace

Direction of S1

001

A A′

∧

∧

110

ab = A′B′ For S2 = tan 11°

311 301 Habit plane pole

a,b before S2 and S1 A,B after S2 A,B after S2 and S1

100

110

Figure 8.7. Stereographic projection showing lattice distortion, S1 , and the shear, S2 involved in the to transformation.

When S1 is applied to a unit sphere, the set of vectors which remain undistorted generate an elliptical cone. The positions of this cone before and after the application of S1 are represented in the stereographic projection shown in Figure 8.7. It can be seen that the superimposition of a certain fraction of the shear S2 can lead to the fulfilment of the invariant plane strain (IPS) condition for a plane defined by the undistorted vectors a and b as shown in Figure 8.7. The pole of this irrational plane is located within the stereographic triangle defined by the ¯ and (301) . This habit plane prediction matches well with poles (311) , (100) that experimentally determined for the -hydride precipitates forming within the -phase matrix in a Zr-20% Nb alloy. The Wechsler, Liebermann and Read (WLR) methodology of martensite crystallography is applied for the → transformation for the precise determination of the habit plane and the magnitude of the lattice invariant shear. The lattice strain S = S1 · S2 represented with reference to the axial system, xi , can be transformed into the fct axial system by employing a similarity transformation:


733

⎡

⎤ 09088 02234 0 0 ⎦ SB = x RB S B Rx −1 = ⎣ 01768 13090 0 0 09734

(8.14)

As mentioned earlier, a superposition of the S2 shear on S1 can generate the -hydride lattice. If the direction of S2 is reversed, a second variant of the -hydride crystal is created. These two -orientations have a twin relation between them. These two shears, denoted as S+2 and S−2 are along the following two directions: ¯ 111 121 ¯ S2+ 110 110 ¯ 111 12 ¯ 1 ¯ S2− 110 110 Consistent with this fact, the lattice-invariant shear (LIS), has been selected to ¯ , the lattice-invariant shear, P with reference to the axial system be (110) [110] defined by the shear direction, d, the shear plane normal, n, and the vector, dxn, can be expressed as: ⎡ ⎤ 1 0 g P = ⎣0 1 0⎦ (8.15) 0 0 1 where, g is the magnitude of LIS. The LIS matrix, PB for a shear along (110) ¯ represented in the bcc axial system, will be [110] ⎡ ⎤ ⎤ 1 1 0 0 0 0 g PB = ⎣ 0 1 0 ⎦ + ⎣ 1¯ 1¯ 0 ⎦ 2 0 0 0 0 0 1 ⎡

(8.16)

The macroscopic shape strain, FB , is then given by F B = PB · S B ⎤ ⎤ ⎡ 05427 05427 0 0 0904 02137 = ⎣ 01814 12991 0 ⎦ + g ⎣ 05427 05427 0 ⎦ 0 0 09734 0 0 0 ⎡

(8.17)

The IPS condition can be met by adjusting the value of g. In the present case, the magnitude of the lattice invariant shear, g, has been found to be 0.1699 and

734


substituting this in Eq. (8.17) the average macroscopic shape strain matrix can be determined to be ⎤ ⎡ 09962 01215 00 (8.18) FB = ⎣ 00892 12069 00 ⎦ 00 00 09734 The strains in three principal directions could be determined by finding the eigen values of the macroscopic strain matrix FB and these are given by: 1 = 09734

2 = 10

and

3 = 12131

where 1 , 2 and 3 are the strains along the three principal axes. It could be thus seen that 1 < 1, 2 = 1 and 3 > 1 which is the necessary and sufficient condition for IPS (Lieberman et al. 1957, Wayman 1964). Therefore, an invariant plane strain condition in → hydride transformation could be achieved when the magnitude of the shear is 0.1699 for the LIS system given by shear S+2 . Furthermore, applying the WLR matrix methodology (Lieberman et al. 1957, Wayman 1964), the habit plane or the interface plane (HB ) for the correspondence variant given in Figure 8.5 and for the LIS given by shear S+2 corresponding to two opposite shear directions are found to be the same: HB + = −00850 29968 10000 and HB − = −00850 29968 10000 Following the method of comptuation described in WLR theory (Lieberman et al. 1957), the other relevant crystallographic parameters for the given shear system were calcualted and the results are presented in Table 8.2. Table 8.2. The computed crystallographic parameters of → transformation, for correspondence variants shown in Figure 8 and first shear system. Crystallographic parameters

Computed values Positive solution

Negative solution ⎤ ⎡

⎡

090376 10001 −00059 −00020 ⎣−00059 12098 00700 ⎦ ⎣ 007084 Shape strain matrix −005749 00025 −00873 09709 00089 −03163 09486 00089 Shape strain shear direction −00262 09230 −03839 00262 Shape strain direction Magnitude of shape strain

0.2398

⎤ −007302 006930 105375 −005101⎦ −004362 104140 −03163 −09486 −09230 −03839 0.2398


735

It is to be noted that the LIS chosen here are the same as S+2 and S−2 and that the LIS magnitude of (0.1699) corresponds to nearly half of the shear S2 . This means that a -plate which consists of two twin related -variants (one corresponding to S2+ and the other to S−2 shear) with the ratio of their thicknesses equal to 1:3 will satisfy the IPS condition. Experimentally, a twin thickness ratio of 1:3 is often observed in -plates containing (111) twins, the twin plane being derived from the parent (110) plane. The habit plane solutions obtained from the WLR analysis have been found to be of the (130) type which also match closely with the experimentally determined habit plane indices. From these observations it is clear that the phenomenological theory of martensite crystallography can be used for the prediction of crystallographic observables such as the habit plane and the magnitude of lattice-invariant shear (or ratio of twin thicknesses) in the context of the precipitation of -hydride plates in the -phase matrix (Srivastava et al. 2005). 8.3.4 Crystallography of → transformation The crystallography of the → transformation and the internal structure of -hydride precipitates in the -matrix have been studied in detail by Weatherly (1981), who has pointed out that there are two distinct orientation relationships and habit planes exhibited by -hydride precipitates in the -matrix. These are grouped as type I and type II, the crystallographic details of which are given in Table 8.3. The hcp -structure can be transformed into the fcc -structure by altering the stacking sequence from the ABABAB to the ABCABCABC type . The lattice strain required for transforming the hcp -lattice to an fcc lattice (an imaginary precursor to the -hydride phase) consists of a small dilation of the (0001) planes (to increase the c/a ratio from 1.593 to the ideal value of 1.633) and a simple shear to change the stacking sequence. This fcc structure (with a lattice parameter of 0.4575 nm can be transformed into the fct -hydride structure (a = 04596 nm and c = 04969 nm by a second homogeneous deformation. The overall ¯ , [1010] ¯ and [0001] is lattice strain, referred to the axes [1210]

Table 8.3. Orientation relationship and habit planes for Type I and Type II -Hydride precipitation in -Zr. -Hydride Type I Type II

Habit plane


¯ } {1010 ¯ } {1017

¯ 0] ; (001) (0001) [110] [121 ¯ [1210] ¯ ; (111) (0001) [110]

736


⎡

⎤ 10048 0 0 10592 03356 ⎦ SH = ⎣ 0 0 −00471 10308

(8.19)

It may be noted here that the / correspondence observed for the type II -hydride precipitates is crystallographically equivalent to that depicted in Figure 8.5. The fact that the lattice strain along one of the principal axes is ¯ direction for less than 0.5% allows one to neglect the strain along the [1210] an approximate analysis which can be reduced to two dimensions. Weatherly (1981) has shown from the construction of the unit circle and of the deformed ellipse that the IPS condition is satisfied for habit planes which are indicated by the undistorted vectors (AB) and after rigid body rotation (A B ) in Figure 8.7 ¯ zone axis. As seen in this construction, two which corresponds to the [1210] ¯ possible habit planes are predicted, both having a normal lying along the [1210] direction. The habit plane normal for the first solution lies approximately 12 from the [0001] pole and a rigid body rotation of 3 is required to bring the undistorted plane back to its original orientation. For the second solution, the ¯ pole and a rigid body rotation habit plane normal lies 16 away from a (1010) ¯ makes an angle of 20 is required. The experimentally observed habit of (1017) of about 14 with (0001) , indicating the validity of martensitic crystallography in the case of -hydride precipitation in -Zr.

¯ -hydride plates of type I which exhibit the 1010 type habit plane have been found to follow one of the two orientation relations listed in Table 8.3. For -plates corresponding to the first orientation relation, (001) (0001) , internal twins are seen along two sets of 110 planes which are obliquely inclined to the habit plane. In contrast, for -plates following the second orientation relation, (111) (0001) , the twin planes are parallel and perpendicular to the habit

plane. ¯ The transformation mechanism associated with -plates exhibiting 10 17

¯ habit planes (type II), involves a simple shear of 30 on a 1010 plane in ¯ 10> ¯ direction. It is, however, interesting to note that the lattice correa < 12 spondences for the -plates of type I and type II could be regarded as crystallographically equivalent if one considered the to transition to occur in two conceptual steps: the first step being the to transition in accordance with the Burgers correspondence followed by the second step comprising the to transition as discussed in the preceeding section. A stereographic projection showing the approximate orientation relations between and through an intermediate orientation brings out the equivalence of two lattice correspondences (Figure 8.7). It could be seen from this stereographic projection that for the type I plates, the (110) plane which is derived from the (0001) plane becomes the


737

(111) plane, whereas for the type II plates, other 011 planes are converted to 111 planes. 8.3.5 Mechanism of the formation of -hydrides The -hydride phase precipitates form in both the - and the -alloys at high quenching rates. Alloys containing hydrogen to a level exceeding the terminal solubility, will invariably produce -hydride precipitates. The charcteristic features of -hydride precipitates, namely, surface relief on polished surfaces, strict orientation relationship and habit plane, operation of the invariant plane strain condition, periodic internal twinning and rapid transformation kinetics suggest that the transformation involves a lattice shear. The fact that the transformation occurs at sufficiently low temperatures where self-diffusion of zirconium (or titanium) atoms is negligible also suggests that the transformation of the - or the -lattice into that of -hydride does not involve diffusive jumps of the metal atoms. However, hydrogen migration which is known to be very fast even at such low temperatures ( th , the threshold stress for reorientation of hydride being th and a migration zone where < th . The processes which operate in these zones are schematically illustrated in Figure 8.12. Experimental observations of the DHC phenomenon have revealed the following important features (Singh et al. 2002a): (1) DHC crack initiation is associated with an incubation period and fracture surface is usually marked with striations (Figure 8.12(b)). (2) The initiation of DHC is associated with a threshold stress intensity factor, KIH , below which the DHC crack growth velocity, VDHC , is too small to be detected; (3) KIH is practically independent of material strength; (4) For a given material and for KIH < KI < KIC , VDHC is independent of the applied stress intensity factor, KI (where KIC is the fracture toughness of the material); (5) VDHC increases with an increase in the strength of the material; (6) VDHC increases with temperature due to increased hydrogen diffusivity and solubility at higher temperatures. Though softening of material causes an increase in the process zone size, the increased hydrogen diffusivity has a dominating effect on VDHC . The variation of VDHC with temperature is reported to exhibit an Arrhenius type relationship, the activation energy for DHC being in the range of 40–70 KJ/mole which corresponds to the sum of the activation energy of hydrogen diffusion (30 kJ/mole) and the enthalpy of mixing of hydrogen in zirconium (38 kJ/mole). 8.4.5 Formation of hydride blisters In the presence of a thermal gradient, hydrogen migrates down the temperature gradient. Once the solid solubility at the localized low temperature spot (hereforth called cold spot) is exceeded, hydride precipitation starts. A sustained thermal gradient allows continuation of hydrogen migration from the high temperature to the low temperature region as a steady state concentration gradient persists under such a condition. The volume fraction of the hydride phase at the cold spot increases to near 100% and a semiellipsoidal module of hydride forms at the cold spot. Since the transformation of zirconium metal into hydride is associated

748


σy = K

σy

√2πr

Stress (σ)

σo

Modified stress distribution elastic + plastic

σth

Distance (r)

rp

Process zone

Reorientation Migration zone zone (a)

80 μm (b)

Figure 8.12. (a) Schematic representation of three zones ahead of a sharp crack. (b) Fracture surface showing striations resulting from stepwise growth of delayed hydride cracks.


749

with an increase in volume, a bulge appears on the surface of the cold spot. This hydride bulge, due to its appearance, is called a hydride blister. The phenomenon of blister formation attracted special attention of the nuclear engineering community after the failure of a zircaloy-2 pressure tube in a pressurized heavy water reactor, PHWR (Pickering - Unit 2) in Canada. In the PHWR design, several hundred pressure tubes are arranged in a lattice in a large vessel called calandria (Figure 8.13). Each of these pressure tubes which contains a series of fuel bundles pass through calandria tubes, the gap between a pressure tube and the surrounding calandria tube is maintained by a set of spacers known as garter springs. Pressure tube carries coolant water at around 300 C whereas the calandria tube remains in contact with moderator water maintained at a temperature of 60–80 C. Pressure tubes have a tendency of sagging because of the deflection due to the fuel load and due to the in-reactor creep. When the sagging of the pressure tube results in the establishment of a contact between a pressure tube and a calandria tube, the contact points become cold spots (as shown in Figure 8.14(a) and (b)) which are susceptible to hydride blister formation.

Figure 8.13. Lattice arrangement of pressure tubes and calandria tubes in pressurized heavy water reactors.

750

Phase Transformations: Titanium and Zirconium Alloys Presure tube

Calendria tube 80°C 300°C

Garter spring (a) Before creep deformation 80°C 300°C

Contact point (b) After creep sagging OD of the pressure tube

Circumferential direction

380 μm

Region III

Radial direction

O

Region II

Region I

(c)

SYY - STRESSES

VIEW : –54.10852 RANGE : 110.4911

110.5

98.73

86.98

75.22

63.46

51.71

39.95

28.19

16.43

4.677

–7.080

–18.84

–30.59

–42.35

–54.11

ROTX 0.0 ROTY 0.0 ROTZ 0.0

EMRC–NISA/DISPLAY

Y X

MAR/05/01 15:46:26

(d)

Figure 8.14. (a) and (b) show pressure tube/calandria tube contact creating cold spot on the power. (c) A section of hydride blister showing both the radial and circumferential hydrides. (d) Computed stress field inside the blister and the matrix surrouding it.


751

The parameters which influence the growth rate of hydride blisters are the difference in temperature between the bulk of the material and the cold spot, thermal conductivity at the contact point and the hydrogen content of the material. Larger the temperature difference and sharper the thermal gradient, higher the blister growth rate. A higher hydrogen level in the bulk of the material no doubt promotes the growth of blisters. Metallographic examination of the section of a hydride blister along the radial-circumferential plane of a pressure tube shows three distinct regions (Figure 8.14(c)), viz. the core region where the hydride volume fraction is nearly 100%, the reoriented hydride region where a large fraction of hydride plates are reoriented from their original habit to directions radial to the semiellipsoidal blister and the region where hydride precipitates retain their original habit. The stress field present around a growing blister is responsible for reorienting hydrides in the region adjacent to the blister (Figure 8.14(d)) (Singh et al. 2003). Two distinct types of blister morphology have been encoutnered, namely (a) a single convex hump; and (b) a nearby circular ring which is made up by joining a series of independently nucleated blisters (Singh et al. 2002b). The former type is formed when all the hydrogen present in the alloy is first brought into solid solution and the cold spot is created subsequently. Under this condition, blister nucleation occurs right at the cold spot. In any case, these blsiters cannot grow beyond a size where their periphery touches the temperature corresponding to the TSSP of hydrides. The latter type (ring morphology) is formed when the cold spot is maintained during the heating-up operation. As a consequence, the hydride precipitates which remain undissolved in the vicinity of the cold spot act as nucleating centres for blister formation. As the hydrogen flux is directed towards the cold spot, hydrogen atoms are arrested at the point where undissolved hydride precipitates are present. The ring of blisters, therefore, mark the temperature contour corresponding to the TSS limit for hydride dissolution under the imposed thermal gradient. In this context it may be emphasized that the TSS limit for precipitation and for dissolution of hydrides in -zirconium are different and are denoted by TSSP and TSSD, respectively. While TSSP is the controlling factor for the single blister morphology, the ring type blister formation is governed by TSSD (Singh et al. 2002b). For the nucleation of blisters, a minimum hydrogen concentration, cBFT , called blister formation threshold is required. However, cBFT is dependent on the TSS, ∗ , at the cold spot temperature. The relationship of these concentrations with the cTSS bulk and the cold spot temperatures can be derived from the steady state condition in absence of stress, where the net flux J of hydrogen is given by Eq. 8.21.

752


This partial differential equation can be reduced to Q∗ 1 dcx + =0 cx dT RT 2

(8.23)

Separating the variables and on integrating between the bulk dissolved hydrogen ∗ at the cold spot temperature, T ∗ we get concentration, cb and cTSS

∗ cTSS

cb

dcx Q∗ T dT =− cx R Tb T 2 ∗

(8.24)

which reduces to Q∗ Tb − T ∗ cBFT = exp − ∗ cTSS R Tb · T ∗

(8.25)

∗ The ratio of the two concentrations cBFT /cTSS , increases with an increase in the ∗ cold spot temperature, T , as is shown in Figure 8.15 where the bulk temperature, Tb is maintained at 573 K (Singh et al. 2002b). Since cTSS depends on the direction of approach of temperature (i.e. TSSP and TSSD), cBFT will also vary depending on whether it is measured during a heating or a cooling operation.

50

0.7

TSSD TSSP c BFT /c TSS

0.6 0.5

30

0.4

20

0.3 0.2

c BFT/c TSS

c TSS (ppm)

40

10 0.1 0

250

0.0

300

350

400

450

500

550

Tcs (K)

Figure 8.15. Hydrogen concentration corresponding to blister formation threshold (cBFT ) as a function of the cold spot temperature, Tcs .


8.5

753

THERMOCHEMICAL PROCESSING OF Ti ALLOYS BY TEMPORARY ALLOYING WITH HYDROGEN

Titanium and most of its commercial alloys have a high solubility for hydrogen, being capable of absorbing upto about 6 at.% hydrogen at 600 C. Since hydrogen entry in titanium alloys is reversible, temporary alloying with hydrogen can be very effectively used in stabilizing the -phase which in turn assists in refinement of the microstructure and in improvement of fabricability of these alloys. Since the alloy chemistry is changed during the processing operation, the technique is known as thermochemical processing. The basic principle of this processing technique is discussed here as an example where hydrogen-induced phase transformations have beneficial effects. After the processing is completed, hydrogen can be extracted from the alloy by vacuum annealing. The concept of constitutional solution treatment can be explained using a pseudo-binary phase diagram for Ti–6Al–4V with hydrogen (Figure 8.16). Since hydrogen stabilizes the -phase, it is possible to solutionize Ti–6Al–4V into a single phase -solid solution by hydriding at a temperature above the eutectoid temperature (Te ) of 800 C. This hydrogenated alloy (containing 0.5–1 wt% H) on cooling transforms into either + TiH or Orthorhombic martensite ( ) depending on the cooling rate. The latter on subsequent ageing decomposes into a mixture of + TiH. The large volume expansion associated with hydride formation results in an accumulation of high density of dislocations in the matrix. When this material is dehydrogenerated, a fine -phase forms by the decomposition of the hydride to a mixture of and spherodized -phase. The low transformation and dehydrogenation temperatures and strain in the matrix are responsible in substantial refinement of the microstructure and consequent improvement in mechanical properties. Hydrogen (at.%)

Temperature (K)

10

20

30

40

β

1273

β+X

α+β 1073 α

X

873

α+X 0.2

0.4

0.6

0.8

1.0

Hydrogen (wt%)

Figure 8.16. Ti-6V-4A1 + H pseudo binary phase diagram.

1.2

754

Phase Transformations: Titanium and Zirconium Alloys Table 8.4. Thermochemical treatment cycles. Designation

Treatment sequences

Constitutional solution treatment -quench followed by hydride-dehydride Hot isostatic pressing or vacuum hot pressing High Temperature hydrogenation

Hydrogenate at T > Te → cool to T < Te → dehydrogenate -solution treatment → water quench → hydrogenate at T < Te and dehydrogenate at T < Te Hydrogenate to increase -volume fraction → hot deformation → dehydrogenate Hydrogenate in → cool to room temperature → dehydrogenate at T < Te

A variety of thermochemical processing cycles, as listed in Table 8.4, have been tried. Froes and Eylon (1990) have summarized the effects of these thermochemical treatments on the microstructure and mechanical properties of titanium alloys.

8.6

HYDROGEN STORAGE IN INTERMETALLIC PHASES

8.6.1 Laves phase compounds Hydrogen storage materials (HSMs) are usually intermetallics containing interstices with a suitable binding energy for hydrogen which allows its absorption or desorption near room temperature and atmospheric pressure. A promising candidate for hydrogen storage is the class of Laves phase compounds having the formula unit of AB2 . Laves phase include fcc (MgCu2 ), hexagonal C14 (MgZn2 ) and di-hexagonal C36 (MgNi2 ) structures. Since they transform from one to another during heating and cooling (typically C14 at high temperatures and C15 at low temperatures), hydrogen absorption and desorption can essentially be considered as a phase transformation process. The C15 structure is an fcc-based structure containing six atoms (two formula units) in the primitive unit cell, while C14 and C36 structures are hexagonal structures containing 12 and 24 atoms in the primitive unit cell, respectively. The interstitial sites occupied by hydrogen in AB2 Laves phases are tetrahedral sites formed by two A and two B atoms (2A2B site) or by one A atom and 3 B atoms (1A3B site) or four B atoms (4B site). There are 17 tetrahedral sites per formula units: twelve 2A2B sites, four 1A3B and one 4B site for both C14 and C15 structures. While in C15 structures, A or B atoms within each type of sites (2A2B, 1A3B and 4B) are locally equivalent, this is not the case for C14 structure. Depending on the local environment of the tetrahedral interstice, 12 2A2B sites and 4 1A3B sites per formula units in the


755 Y

Active

X

Z

(a)

5

4 3 7

2

8 6 9 1 (b)

(c)

Figure 8.17. (a) B2 crystal structure, (b) Crystal structure of Laves phase compounds C14 (hexagonal) open circles stand for A atoms and solid circles for B atoms and (c) the C15 Laves structure, where the larger and the smaller circles represent A (Zr) and B (X = V, Cr, Mn, Fe, Co, Ni) atoms, respectively. Here, for example, a 2A2B site is formed by the atoms labeled 2, 4, 7 and 8; a 1A3B site by atoms 1, 6, 8 and 9; and a 4B site by atoms 6, 7, 8 and 9.

C14 structures are further sub-grouped into six 2A2B (I) sites, three 2A2B(k2 ) sites, 1.5 2A2B(h1 ) sites, 1.5 2A2B(h2 ), one 1A3B(f) site and three 1A3B (k1 ) sites. To depict the interstitial sites formed by A and/or B atoms, a ball stick model of the fcc structure is given in Figure 8.17. The 2A2B sites has the largest interstitial hole size and the 4B site has the smallest hole size (Hong and Fu 2002). The volume change associated with hydrogen absorption could be as large as +20%. Such a large volume change during hydrogen absorption and desorption results in cyclic loading and unloading of the matrix which may lead to disintegration of the host lattice.

756


8.6.2 Thermodynamics Most of the thermodynamic properties of a HSM can be obtained from pressure– composition isotherms (PCI). Figure 8.18(a) illustrates a typical PCI diagram (Sinha et al. 1985). The points of intersections of mildly sloping plateau with the steeply sloping portion of an isotherm define the phase boundaries / + and ( + /, where is the solid solution of hydrogen in the host alloy and is the hydride of the alloy. The plateau (mildly sloping region) pressure for formation of hydrides (Pf ) is higher than that corresponding to dissociation of hydrides (Pd ) due to hysteresis effect. The degree of hysteresis is expressed by the logarithmic function (1/2) RT 1n Pf /Pd which is the free energy loss per mole of atomic hydrogen in completing a hysteresis loop. With increase in temperature, plateau pressure increases. Equilibrium pressures in the plateau region and its slope are important in deciding the application of HSM. Using Vant Hoff equations: ln Pi =

Hi Si − RT T

where i stands for f (formation) or d (dissociation) of hydrides, the enthalpy of formation and dissociation can be obtained from PCI by plotting plateau pressure against the inverse of absolute temperature. The negative of the slope of such a plot will yield enthalpy change value and the intercept will yield the change in entropy values. 8.6.3 Ti- and Zr-based hydrogen storage materials In general, a good HSM for automobile applications should have dissociation temperature and corresponding equilibrium vapour pressure comparable to that of the ambient. Most of the developed hydrogen storage alloys can be classified in two groups, viz. AB5 and AB2 types. LaNi5 is a typical example of the AB5 type hydrogen storage alloys, which is mainly used in Ni-metal hydride batteries. Hydrogen storage capacity of AB5 alloys is about 1.5 wt% (Taizhong et al. 2004). In the AB2 type alloy, A = Ti or Zr and B = Ni, V, Mn Cr etc. These alloys usually possess the structure of Laves phase. The AB2 type Laves phase alloys have been regarded as promising HSMs because of high hydrogen storage capacity. Both Ti- and Zr-based AB2 have been explored for applications as HSM and are briefly discussed in the next sections. Zr-based Laves phase alloy hydrides are very stable with very low dissociation pressure for hydrogen and hence are not suitable for automobile application. On the other hand, Ti-based Laves phase alloy hydrides are unstable but their plateau characteristics are not suitable for automobile applications. Thus for both Ti- and Zr-based AB2 type alloy, substitution of both A- and B-type element are being used to modify the plateau pressure, its slope and hysteresis to suitable values.


757

30°C ZrCr1 – γ Fe1 + γ

Pressure (Nm)

0.5 0.3 0.2

0

1

2

3

(a)

0.6

Composition (HM)

Ti0.99 Zr0.01 Fe

Ti0.8 Zr0.2 Fe

0.5

Ti0.9 Zr0.1 Fe

Ti0.9 Zr0.1 Fe (No H.T.)

0.4 0.3

0.2

0.1

0.01

TiFe 0.1

1

10

100

1000

Time (min) (b)

Figure 8.18. (a) Typical pressure–composition isotherms for ternary Zr–Cr–Fe alloys. (b) Influence of Zr addition on the hydrogen absorption curves for TiFe alloys.

758


8.6.3.1 Ti-based hydrogen storage materials Among the Ti-based systems, TiFe, TiNi2 , TiNi, TiMn2 , TiCr and TiV intermetallic compounds were investigated for hydrogen storage and purification. One of the significant problems associated with TiFe was the difficulty in activation. Substitution of Nb or Ta or V for Fe and substitution of Nb or Cu for Ti in TiFe is reported to accelerate activation. The TiFe alloys, to which Fe2 O3 with Cu were added for Ti, comprised of TiFe matrix and precipitates of Fe2 Ti and the oxide (Fe7 Ti10 O3 ) phases. The interface between the matrix and the precipitates were suggested to be active sites for the hydriding reaction and as entrance sites for hydrogen to diffuse into the alloy. Addition of small quantities of Zr and Nb enhances the ease of activation (Figure 8.18(b)). TiMn alloys have been investigated due to their easy activation, good hydriding– dehydriding kinetics, high hydrogen storage capacity and relatively low cost. TiMnx (with x = 1 to 2) have C14 type hexagonal crystal structure. The hydrogen storage capacity of Ti–Mn hydrogen storage alloy was reported to be 1.5–1.8 wt%. There are two major categories of Ti–Mn HSM, i.e. TiMn2 and TiMn15 . The main problem with the Ti–Mn alloys is their high equilibrium plateau pressure, which limits their practical application. Another shortcoming of these alloys is that hydrides in these alloys are characterized by large hysteresis effects. V is reported to reduce the hysteresis. Homogenization treatment helps in obtaining clear plateau pressure. Several studies have focused on the roles of solid–gas and electrochemical reactions. The activation of the alloys plays a key role in hydrogen absorption process, since it defines the reaction rate of the hydrogen with the metal and its incorporation in its structure. The hydrogen storage capacity of HSM is a function of the crystal structure–lattice parameter and void size. By suitable alloying addition, void size can be manipulated resulting in increase in hydrogen storage capacity. In late nineties, it was reported that TiV with a bcc structure absorbs more hydrogen than conventional intermetallic compounds. Zr substitution in Ti–Cr–V system decreases the equilibrium pressure and the hysteresis of the PCI. However, it increased the slope of PC isotherm and reduced the hydrogen storage capacity by forming ZrCr2 . By selecting suitable composition range in the quarternary Ti–Zr–Cr–V system, formation of this phase can be avoided (Figure 8.19). The atomic radius of zirconium (1.62 Å) is larger than that of Ti (1.47 Å) and its electronegativity (1.4) is smaller than that of Ti (1.6). Generally, a metal hydride becomes stable with increasing difference between the electronegativities of hydrogen (2.1) and the metal atom. By adding Zr, therefore, a decrease of the hysteresis and the plateau pressure, and an increase in the width of the plateau and the hydrogen storage capacity due to an increase in lattice volume is expected. Under heat treated condition, the hydrogen storage capacity of Ti016 Zr005 Cr022 V057 is reported to be 3.55 wt% (Table 8.5).


759

Ti

at.%

Cr

at.%

at.%

V

Figure 8.19. Composition range showing the largest effective hydrogen storage capacity in the Ti–Cr–V alloy system.

Table 8.5. Hydrogen storage capacities of the composition controlled and heat treated Ti–Zr–Cr–V alloys. Composition

Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026 Ti012 Zr005 Cr045 V026

H-storage capacity

Effective H-storage capacity

H/M

wt%

H/M

wt%

1.50 1.44 1.50 1.73 1.67 1.57 1.71 1.87

2.90 2.74 2.87 3.34 3.18 2.99 3.24 3.35

0.85 0.77 0.79 1.07 0.93 0.86 1.02 1.13

1.65 1.47 1.51 2.07 1.77 1.63 1.92 2.14

8.6.3.2 Zr-based hydrogen storage materials Various studies performed so far on Zr-based AB2 type HSM have been directed to increasing the equilibrium hydrogen vapour pressure in the Zr-based Laves phase alloys without markedly reducing the absorption capacity by partial substitution of A or B type elements by other elements. The A element can be substituted by Ti while the B element by Cr, Mn, Fe, Co, Ni and Cu. The reduction in hydride stability is possible due to any or all of the following effects: the reduction in the size of the hydrogen occupation site, the change in chemical affinity for hydrogen and the electronic factor. The reduction in binding energy of hydrogen in C15 Zr X2 ( X = V, Cr, Mn, Fe, Co, Ni) is shown in Figure 8.20. Most

760


Binding energy of H (kJ/mol H)

30

0

–30

2A2B site 1A3B site

–60

4B site

–90

V

Cr

Mn

Fe

Co

Ni

Figure 8.20. Binding energy of hydrogen in C15 ZrX2 (X = V, Cr, Mn, Co, Ni).

research has been concentrated on the ZrMn2 type and ZrCr2 type pseudobinaries. Elemental substitution in pseudo-binaries can be classified according to the following: (1) A element substitution, primarily with Ti: e.g. Zr1−x Tix Cr2 and Zr1−x Tix Mn2 . (2) B element substitution with a different transition element: e.g. Zr(Mn1−x (Fex )2 , Zr(Cr1−x Fex )2 . (3) Hyperstoichiometric substitution of B element: e.g. ZrMn2 + x, ZrMn2 Fex , ZrMn1 + xFe1+y (4) Any combination of the above: e.g. Zr1−x Tix MnFe and Zr1−x Tix (Fe1−y Mny )z alloys. The hydrogen absorption characteristics of Zr-based stoichiometric AB2 HSM such as ZrMn2 , ZrV2 , ZrFe2 , ZrCo2 and ZrCr2 were first studied by Shaltiel’s group (Shaltiel et al. 1977). The hydrides of the first two alloys were observed to be too stable to be of practical significance. The stoichiometric ZrMn2 hydride, for example, exhibits a dissociation pressure of 0.007 atm at 323 K. ZrMn2 crystallizes in the C14 (MgZn2 ) structure. The same structure is observed for ZrMn2 alloys in which Zr has been partially substituted by Ti and Mn by Fe. Since the hydride in ZrMn2 is stabilized primarily by interaction of hydrogen with Zr, it was suggested that the vapour pressure of hydride might be increased by having a system that was


761

substoichiometric in Zr. Investigation of ZrMn2+x binaries, ZrMn2 Fex ternaries and a series of related systems have established that the dissociation pressure and the hydrogen capacity of these alloys depend on the Zr:Mn:Fe ratio in the lattice and the dissociation pressure was manifold higher than ZrMn2 without significantly reducing the hydrogen storage capacity (Sinha et al. 1985). Though these alloys were found to absorb large quantities of hydrogen, the hydrides formed were too stable to be of practical significance. ZrMn1+x Fe1+y , where Zr is partially substituted by Mn and/or Fe and Zr1−x Tix MnFe, where Zr is partially substituted by Ti are reported to be more attractive than LaNi5 because of favourable combination of properties such as lower cost, rapid kinetics of hydrogen sorption and lower endothermal nature of dehydrogenation. In comparison with Mn and Fe, the hyperstoichiometric Ni and Co is reported to be more effective in augmenting the decomposition pressure of ZrMn2 hydride, although their presence reduced hydrogen storage capacity of the alloy. Sinha et al. (1985) reported that the multicomponent Zr-based alloy having C14 structure appear to be very promising HSM. The ZrCrFe1+x alloys, in particular, have many attractive properties. The vapour pressure of ZrCr2 hydride is augmented by several orders of magnitude without significantly reducing its hydrogen storage capacity when 50% Cr is replaced by Fe and in addition, hyperstoichiometric Fe is present in the lattice. The ZrCrFe15 alloy, for example, exhibits a dissociation pressure of its hydride of about 2 atm at 23 C. The hydrogen storage capacity of this alloy is comparable to that of LaNi5 . The effectiveness of alloying elements in destabilizing ZrMn2 hydride is reported to be in the following order: Cr < Mn < Fe < Ni < Co. Sinha et al. (1982) reported that 20% substitution of Zr by Ti raises the equilibrium pressure five fold in (Zr1−x Tix )Mn2 . The change in PCIs for hydrogen absorption and desorption in ternary ZrCr1−y Fe1+y alloy is shown in Figure 8.21 with increasing Fe content (Park and Lee 1990). Substitution of iron by chromium raises the two phase plateau pressures for hydride formation and dissociation. The extent of hysteresis was also observed to increase. The overall trend in PCIs of Zr09 Ti01 Cr1−y Fe1+y for the ranges 0 < y < 0.4 is shown in Figure 8.22 (Park and Lee 1990). Unlike Ti-free ZrCr1−y Fe1+y alloy, well-defined plateaux with low slopes are formed in the two phase region for the whole composition range y. It is reported that 10% substitution of Zr by Ti in ZrCr1−y Fe1+y alloy yield the best of the Laves phase alloy for various applications due to their low hysteresis and low slopes. 8.6.4 Applications In the recent years, hydrogen has received worldwide attention as an alternative energy carrier which can substitute petroleum in internal combustion engines. The possible applications of HSMs are in automobile industry, moderators in nuclear

762

Phase Transformations: Titanium and Zirconium Alloys 30°C Zr0.8 Ti0.2 Cr1 – y Fe1 + y

30°C Zr0.9 Ti0.1 Cr1 – y Fe1 + y Y

Y

0.4 0.3 0.2 0.1 0

10

PH2 (atm)

PH2 (atm)

10

0.4 0.2 0

1

1

0.1

0.1 0

2

1

H/AB2 (a)

3

0

2

1

3

H/AB2 (b)

Figure 8.21. Pressure–composition isotherms for Zr–Cr–Fe–H2 systems in which Zr has been partially replaced by (a) 10% Ti and (b) 20% Ti at 303 K.

reactors, heat pumps, refrigerators, getters, etc. With the development of fuel cell technology and the widening of hydrogen application fields, methodology of hydrogen storage has attracted more and more attention. Three systems of interest are glass microspheres, liquid hydrogen and metal hydrides. The disadvantages for glass microspheres are that charging requires high hydrogen pressure and temperatures (573 K) while for discharging the microspheres have to be heated to 473 K. On the other hand, liquid hydrogen requires complex handling techniques, a safe cryogenic container and involves cost of liquefaction. For a refrigeration cycle having 33% efficiency for refrigeration, the refrigeration will require 10 KWh/kg of hydrogen, which accounts for nearly 25% of heat of combustion of hydrogen. In addition to several advantages, the unique feature of metal hydride is the equilibrium existing between the hydrogen and the metal. On cooling, hydrogen is reabsorbed into the metal, in contrast to two other systems in which, hydrogen once librated, remains in the gas phase. However, several of their properties are insufficient to preclude any large scale engineering applications, e.g. hydrides can be contaminated by oxygen, nitrogen, water vapour and carbon-dioxide, all of which lead to loss of hydrogen storage capacity. The driving force for current


0

10

20

763

Oxygen (at.%) 40 50

30

60

70

{

2473

Tin O2n-1

2273 ~2158 K

2073

2143 K

2115 K

1993 1943 (β-Ti)

γ-TiO

β-Ti2O3

1873

~1523 K

1473 β-TiO

(α-Ti) 1273 1155 K

β-Ti1–xO

α-TiO

Ti3O2

1073 873

TiO2 (rutile)

1673

higher magneli phases

Temperature (K)

L

α-Ti1–xO

Ti2O Ti3O

673 5

0

10

15

Ti

25

20

30

35

40

45

Oxygen (wt%)

(a) Oxygen (at.%) 0

10

20

30

40

50

60

70

L+G

3073 2873

L

~2650 K

2673

γZrO2–x 5.5%, 2403 K

2473

Temperature (K)

2273 2073

~2983 K

~2338 K 1.9 2243 4.1 2.0 2128

22

8.6 10.5

β(Zr) 1873

~1798 K 7.4

1673

23.5 βZrO2–x

~1478 K 1473 (α'Zr)

6.9 ~1243 K

1273 6.7

1136 K (α3"Zr)

1073 873

αZrO2–x

(α2"Zr)

~773 K 26.0

6.6 (α4"Zr )

(α1"Zr)

673 473 0

5

10

15

20

Oxygen (wt%)

(b)

Figure 8.22. (a) Ti–O and (b) Zr–O binary alloy phase diagrams.

25

30

35

764


research on HSM is to attain weight reduction, cost reduction, optimizing temperatures, heat transfer to and from hydride bed, kinetics of hydrogen sorbtion and other parameters. As an example, the requirement for the fuel storage system of the hydrogen automobile are (Sinha et al. 1982) listed below: (1) A minimum delivery rate of 1.25 kg/h to meet the suburban driving cycle defined by the Society of Automobile Engineers; (2) A high energy density (energy per unit mass); (3) A low storage volume such that it can be accommodated in a car; (4) A high degree of safety during charging and discharging; (5) A reasonable charging time; (6) No necessity of an external energy source to start-up; and (7) An energy as small as possible for hydrogen retrieval. Hydrogen storage in metals is of special interest, since certain metals, alloys and intermetallic compounds react rapidly with hydrogen in a reversible manner at nearly ambient temperatures and pressures. The safety of the hydride tanks are comparable to that of petrol tanks, which is widely accepted in the community. The energy to weight ratio of the metal hydrides are about five to ten times greater than that of lead-acid battery, although they are still 10–20 times less than the energy to weight ratio of petrol. The important properties of hydrides for aforementioned applications are hydrogen storage capacity, dissociation temperature, vapour pressure of dissociation, ease of activation (is a function of temperature and pressure at which absorption and desorption begins), hysteresis and enthalpy of absorption and desorption.

8.7

OXYGEN ORDERING IN -ALLOYS

8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O The oxygen atoms dissolved in -Ti and Zr occupy the octahedral interstitial sites of the hcp host lattice. These interstitial sites are surrounded by six metal atoms as shown in Figure 8.2. Dagerhamn (1961) have first reported that the incorporation of oxygen atoms in -Ti and Zr causes a somewhat peculiar variation of the lattice parameters with oxygen content and oxygen ordering occuring at and near by the stoichiometric compositions of O/Ti = 13 (Holmbeog 1962) and O/Zr = 21 (Hirabayashi et al. 1974). Taking into account the formation of interstitially ordered superstructures, the partial phase diagrams of the Ti–O and the Zr–O systems have been constructed (Jostons and McDougall 1970, Hirabayashi et al. 1974) which


765

have been incorporated in the recently compiled binary phase diagrams (Massalski 1992) as shown in Figure 8.22. The ordered hcp phases in Ti–O and Zr–O systems are present in wide composition ranges and are produced by second-order ordering of interstitial oxygen atoms. The octahedral holes in the hcp lattice form a simple hexagonal lattice with a = ao and c = co /2 where ao and co are the lattice parameters of the hcp metal lattice which remains unchanged through the ordering transformation. The ordered structures so produced can be described as a stacking of interstice layer normal to the c-axis on which oxygen atoms are distributed. A single interstitial layer √ can be conveniently described in terms of a hexagonal net with translation of 3ao , as shown in Figure 8.2(c). Oxygen atoms can occupy A, B and C positions which have their coordinates at (0,0), ( 23 , 13 ) and ( 13 , 23 ), respectively. If only A positions are filled by oxygen atoms, the plane is designated as A plane, If A and B positions are filled, the plane is (AB) and when all A, B and C positions are filled, the interstitial plane is indicated as (ABC). Vacant oxygen layers are denoted by (V ). The arrangement of oxygen occupation is not identical in the Ti–O and the Zr–O systems. While in the Zr–O system oxygen atoms in fully ordered state are distributed regularly in every layer of interstice plane with the spacing of co /2, those in the Ti–O system are positioned in every second layer with the spacing co . This seems to be reasonable due to a larger spacing for Zr (co = 0515 nm) compared to that of Ti (co = 0468 nm). As the oxygen concentration changes, the layerwise distribution of oxygen interstitials alters and a series of closely related structures are produced. For the Ti–O system, the oxygen arrangements in Ti6 O, Ti3 O and Ti2 O can be described by the following sequence of interstital and vacant layers: Ti6 O : AVBV, Ti3 O : ACVBCV, Ti2 O : ABCVABCV where V denotes a vacant interstice plane and A, B, C are oxygen interstice planes as described earlier. These structures, as schematically illustrated in Figure 8.23, are closely related to each other and the structural change can be viewed to occur continuously as a function of composition. This point can be illustrated by plotting the occupation probabilities, p, q and r as a function of composition, where p, q and r are defined for the sites A at Z = 0 and B at Z = 21 , the sites C at Z = 0 and 21 , and the sites A at Z = 21 and Z = 0, respectively (Yamaguchi et al. 1970). This shows that the oxygen distribution in the compoO sition range, 16 ≤ Ti ≤ 13 , the structure changes by continuous filling of C sites by oxygen atoms in the ACVBCV layered sequence. The oxygen ordering in the Zr–O system is somewhat complicated. The stacking O sequences of the oxygen layers are basically expressed as ABC at Zr ≤ 13 O 1 and ACBACB at Zr ≤ 2 . The former is isomorphous with -Ni3 C, being

766


√3 ao C

Co

B

α'

2Co

B

(d)

A

A (a) Ti6O

(b) Ti3O

(c) Ti2O

Figure 8.23. Schematic illustration of packing sequence of various Ti-oxides.

called Zr3 O1−x or ZrOx type. The latter is denoted as the ZrO2 type, being a derivative structure of -Fe2 N in which the occupation probabilities, for the sites A, B and C are given as 1 ≥ pA pB > pC . Another type of superstructures O ≥ 13 . This structure known as ZrO with the sequence ABAB is found at Zr is isomorphic with Ni3 N. In addition to the three superstructures, a series of long period structures with various types of stacking sequences are encountered near the O = 13 . Hirabayashi et al. (1974) have listed the long stoichiometric composition Zr period stacking structures of interstially ordered Zr–O alloys at compositions near O = 13 . It is observed that with an increase in the oxygen content, the probability Zr of hexagonal stacking increases continuously. The preference for ‘hexagonal’ O ≥ 13 can be explained in terms stacking in hyperstoichiometric compositions, Zr of strain ordering. If the ‘cubic’ stacking of the oxygen layers were extended over the hyperstoichiometric region, the nearest neighbour pairs of oxygen atoms along the c-axis with the distance c2o would inevitably be created by introduction of excess oxygen atoms in any vacant site. On the other hand, the hexagonal stacking can contain the excess atoms without forming the closest pairs upto the O = 21 , since the same kind of octahedral holes remain unoccupied composition Zr O at the stoichiometric composition Zr = 13 (in the (A) (B) (A) (B) stacking, C sites remaining vacant everywhere. The nearest neighbour distances of the oxygen atom pairs along the c-axis are 3co and co for cubic and hexagonal stackings, respectively, in the Zr–O system. 2 The probability of finding oxygen pairs with interatomic distance of co increases O ratio as shown in Figure 8.24. with the Zr


Probability of O–O pairs

1/3

1/2 1.0

0.8

0.8 Co

0.6

0.6

Zr–O

0.4

0.4

0.2

0.2

0.0

0.1

0.2

0.3

0.4

Probability of h stacking

1/6

1.0

767

0.5

Composition O/Zr

Figure 8.24. Plot of probability of finding Oxygen atom pairs with interatomic distance of co versus O ratio. Zr

The order–disorder transformation in the Ti–O and the Zr–O systems occur in two steps, the disordered -phase first orders into the -phase in which oxygen atoms are distributed in a regular array of interstitial planes (interplanar ordering), but oxygen atoms within the plane remain randomly distributed. Since oxygen interstitial planes are placed on every second layer in the Ti–O system and on every layer in the Zr–O system, the -phase in these two cases are structurally not the same. The second ordering steps results in a periodic distribution of oxygen atoms within these planes (intraplanar ordering). The sequence of transformations, i.e. → can be shown in the order parameter versus temperature plot for a O Ti–O alloy ( Ti = 032), (Figure 8.25) which shows that the interplanar disordering, → occurs at T2 while the intraplanar disordering is complete at T1 . O The disordering process of the Zr–O alloy ( Zr = 13 ) involves changing of stacking sequences of long period superstructures (diminishing periodicity with increasing temperature) and randomization of the ordered arrangement of oxygen atoms on interstitial planes. The overall transformation, → remains to be of the secondorder type, the order parameter continuously dropping as the transition temperature (Tc ) is approached. The domain structure formed due to oxygen ordering has been found to be similar to the B2 ordering (Hirabayashi et al. 1974). This is because only two antiphase domain configurations as shown in Figure 8.26 are possible in the Ti–O system depending on whether oxygen atoms occupy either the (000) plane or the (00 21 ) plane. At the antiphase boundaries which have been drawn parallel and normal to the c-axis (in Figure 8.26) for simplicity, the interstitial oxygen

768


Degree of order

1.0

0.8

SΙΙ

0.6

SΙ

0.4

0.2

0

T1

T2

∨

∨

O/Ti = 0.32

200

400

600

700

Temperature (K)

Figure 8.25. Order parameters versus temperature plot for

↓

APB

↓

O Ti

= 032.

← Ti ←O

↑ C

Figure 8.26. The domain structure formed due to oxygen ordering in Ti–O system.

layers are displaced by a vector c/2 keeping the metal lattice identical across the boundaries. The phenomenon of oxygen ordering is known to have strong influences on several properties such as hardness, electrical resistivity, thermoelectromotive force and magnetic susceptibility, an account of which is summarized by Hirabayashi et al. (1974) As the oxygen level increases towards the equiatomic TiO composition, the -phase appears in the microstructure. The -phase refers to the NaCl type structure which exists over a wide range of compositons including the stoichiometric


769

TiO. While in the - and the -structures titanium atoms are placed in a hcp lattice, the -structure can be regarded as an fcc arrangement of titanium atoms in which oxygen atoms occupy octahedral interstitial positions with high equilibrium concentrations of vacancies in each sublattice. The → phase transformation, studied by Jostsons and McDougall (1970), has shown several interesting features such as strict adherence to the orientation relation and habit plane, shape change and close relation between the two structures. The usual hcp/fcc rela¯ [110] ¯ has been observed between the and the tion, (0001) (111) ; [1120] -phase. Chill cast samples of Ti–O (35–44.5 at.% O) show grains of parallel plates in the -matrix. The → transformation produces martensitic type surface relief effect. The observed transformation characteristics are consistent with the fact that the two lattices are fully coherent at their interfaces. Specimen of Ti–O alloys ( 31–34 at.% O) can be heated from the -phase to the + phase field at temperatures above 1450 C. A heating and cooling cycle in which the / + transus is crossed results in the formation of -plates in the -matrix during heating and subsequent → → transformation during cooling. 8.7.2 Oxidation kinetics and mechanism Two forms of oxidation have been recognized for zirconium alloys. These are uniform and localized forms of oxidation. Under most conditions of temperature and environment, oxidation of zirconium alloys by coolant water or steam results in growth of uniform oxide films, especially in early stages. However, in some isolated regimes of temperature and environment, viz. in 300 C boiling or oxygenated water and in high temperature (>450 C) and high pressure (>5 MPa), a local form of corrosion occurs. The growth of oxide layer as a result of the corrosion reaction between zirconium alloys and water can be described in terms of two distinct stages, usually known as pre- and post-transition stages. The pre-transition stage is characterized by a decreasing rate of weight gain which corresponds to cubic or quadratic kinetics relation between the weight gain and time (Figure 8.27) expressed in terms of effective full power days (EFPD) of operation of a nuclear reactor. The deviation from usual parabolic growth kinetics in this stage is attributed to the fact that the rate controlling process of the diffusion of oxygen ions through the oxide layer is not through lattice diffusion but through grain-boundary diffusion. The oxide layer grows in the early stages maintaining epitaxy. The ratio of the oxide volume to that of the parent (Pilling–Bedworth ratio) is 1.56 for the Zr/ZrO2 system. Thus, as the oxide grows, the stress build-up due to the volume expansion accompanying oxidation induces a fibrous texture of oxide crystal, which minimizes the compressive stress in the plane of the oxide layer. The presence of the compressive

770


70

Zr-2 M pick-up

50

60 50

40 40 30

Break away

30 Zr–2.5 Nb oxidation

20

20

Max. H pickup (mg dm–2)

Maximum oxide thickness (μm)

Zr-2 oxidation 60

10

10 H pick-up Zr–2.5 Nb 1

2

3

4

5

Time (EFPD) × 1000

Figure 8.27. Long term in-reactor, oxidation and hydrogen pick-up behaviour of zircaloy-2 and Zr–2.5 Nb pressure tubes, showing parabolic and then accelerated linear oxidation and hydrogen kinetics in zircaloy-2. A low and uniform rate of corrosion and hydrogen pick-up is seen in Zr–2.5 Nb alloy.

stress is also a factor responsible for the stabilization of the tetragonal phase in the ZrO2 layer. An examination of the metal/oxide interface reveals that a thin layer (∼10 nm thick) of amorphous oxide is present right on the metal surface. Fine crystallites of zirconia are distributed in the amorphous matrix. The volume fraction and size of the crystallites increase with the distance from the metal surface. The growth of the oxide crystals finally results in the formation of a columnar structure. The intercrystalline boundaries provide the diffusion path of O−2 ions from the environment to the reaction front at the metal/oxide boundary. The oxidation process and the nature of the oxide layer on a zirconium alloy sample are schematically shown in Figure 8.28 As the oxide layer grows, the compressive stress at the outer layer of oxide is not sustained and consequently the tetragonal phase becomes unstable and transforms into the monoclinic phase. Such a transformation causes the formation of a fine interconnected porosity in the oxide film which allows the oxidizing water to come in contact with the metal surface. With the development of an equilibrium pore and crack structure in the oxide layer, the oxidation rate becomes effectively linear, a characteristic feature of post-transition oxidation behaviour. Alloying elements, particularly tin, niobium, and iron present in the -solid solution strongly influence both the kinetics and the mechanism of oxide growth in zirconium alloys.


771

Electrons travel by surface conduction Iron oxide

O2 + 4e– → 2O2–

Monoclinic ZrO2

Intermetallic precipitate (Zr – Fe – Ni)

{

2-Diffusion O along

Columnar oxide

Amorphous oxide

→

≈2 μm

Thin ZrO2 layer

crystallite boundaries

ZrO2 crystals Zr

+202–

→ ZrO2 + 4e

Metallic conduction

Zircaloy - 2

Figure 8.28. Schematic diagram showing the mechanism of the oxidation process and the oxide film structure on zircaloy.

Non-uniform oxide formation, usually referred to as nodular corrosion, is the limiting design consideration in boiling water reactors (BWR). Several mechanisms have been proposed for nodule nucleation which can occur in various sites such as -Zr grain-boundaries and in areas at which the continuous dense oxide layer ruptures in the initial stages due to the presence of large intermetallic precipitates. The best microstructure for resisting nodular corrosion in the operating condition of a BWR consists of a distribution of fine intermetallic precipitates (dia 16 , second-nearest neighbour interaction comes into play rendering Ti6 N phase unstable. With increasing N concentration, filling of alternate B and C sites in alternate layers bring back the dominant first neighbour interactions and the nitrogen activity in the ordered solution is lowered once again. Thus we get a minimum at x = 13 in which case the order is again partially restored. Nitrogen atoms in a hexagonal titanium lattice have been found to distort the lattice as shown in Figure 8.32. The distortion of the host lattice can be resolved into two components, one along the c-direction and the other one in direction perpendicular to it. The c-component of the distortion results in a monotonic increase of the lattice parameter c (Figure 8.33) whereas the distortion component along the basal plane periodically distorts the titanium atoms towards the central nitrogen atom (Figure 8.32). In other words, a nitrogen atom serves as a contraction centre as far as the basal plane is concerned; while it serves as an expansion centre along the c-direction. This is brone out by the lattice parameters “a” and “c” variation with composition, as depicted in Figure 8.33. The c-axis exhibits a monotonic increase with nitrogen content while the lattice parameter “a” initially increases and then reaches a saturation value. Associated with such a local distortion, there is an elastic stress field around each interstitial atom which can drive a martensitic transformation of the ordered hexagonal structure to a tetragonal one. The volume change associated with the structural change in the case of the Ti–N system is small enough to be accommodated by a lattice shear. It is also clear that any such C

A

B

C

Ti C

N

Figure 8.32. Schematic illustration of the second neighbour active interaction. Note that it is mediated by the intervening metal atoms and that an interstitial atom serves as a contraction centre in the basal plane. The arrows point to the direction of displacement of the metal atoms.

778

Phase Transformations: Titanium and Zirconium Alloys N, wt. % 5

10

15

C /a

1.62 C/a 1.58

C,Å

4.80 4.76 C 4.72 4.68

a,Å

2.96

a

2.95 2.94 0.0

0.05

0.11

0.176

x

Figure 8.33. Lattice parameter variation with composition for Ti–N system.

homogeneous crystal lattice distortion must preserve the ordered configuraiton of nitrogen atoms inherited from the parent lattice. Sundararaman et al. (1980) have experimentally shown the existence of a martensitic transformation as one of the precursor steps in the precipitation process of Ti2 N in Ti–N alloys. The salient features of the precipitation reaction as experimentally observed by Sundararaman et al. (1980, 1983) are briefly presented with a view to augment the theoretical support concerning the martensitic transformation: (a) There exists a specific orientational relation between the -phase and the TiNx ¯ 011TiN 1 < x < 1 ; 1210 ¯ 011 ¯ TiN ; phase. (1010 2 x3 x (b) The TiNx precipitates always from on a definite habit plane; (c) The transformation is very rapid; it initiates during quenching; (d) The volume change associated with the transforamtion TiNx is small; = (e) The principal strains along three mutually perpendicular directions are: 1210 ¯ 488%; 0001 = 522%; 1010 ¯ = 128%


779

The above values nearly satisfy the requirement of an invariant plane strain condition for a martensitic transformation. The magnitude of lattice invariant shear is, therefore, quite small. The above mentioned lattice correspondence is consistent with the observed orientation relation between the and the TiNx phase (Figure 8.34). The interaction of strain fields in the -matrix, results in the formation of the observed mottled contrast (Figure 8.35(a)). The induction of the lattice transformation of the host lattice (hcp Ti) and the interstitial ordering of N atoms provides yet another example of a coupled displacive–diffusional (interstitial) transformation. (0001)α (100)Ti2N (100)Ti2N (1010)α ||(011)Ti2N

(010.378)Ti2N [1210]α [011]Ti2N

Figure 8.34. Schematic representation of orientation relation between -(hcp) ordered solid solution ¯ 011Ti N , [1210 ¯ 011Ti N . and tetragonal Ti2 N. The orientational relation is (1010 2 2

3]

[01

0]

[10 0.2 μm

(a)

2.5 nm

(b)

Figure 8.35. (a) Quenched microstructure exhibiting mottled or tweed contrast typical of the initial decomposition stage of supersaturated Ti–N solid solution. (b) High resolution structure image showing the presence of and (isostructural with the Ti2 N phase) regions separated by coherent interfaces.

780


REFERENCES Balasubramanian, K. and Kirkaldy, J.S. (1985) Calphad, 9, 103. Banerjee, D. and Arunachalam, V.S. (1981) Acta Metall., 29, 1685. Banerjee, D. and Williams, J.C. (1983) Scr. Metall., 17, 1125. Dagerhamn, T. (1961) Acta Chem. Scand., 15, 214. Darby, M.I., Read, M.N. and Taylor, K.N.R. (1978) Phys. Status Solidi (a), 50, 203. Dey, G.K. and Banerjee, S. (1984) J. Nucl. Mater., 124, 219. Dey, G.K., Banerjee, S. and Mukhopadhyay, P. (1982) J. Phys. C4, 43. Dutton, R. (1976) Metall. Soc. CIM, 16, 16. Flewitt, P.E., Ash, P.J. and Crocker, A.G. (1976) Acta Metall., 24, 669. Froes, F.H. and Eylon, D. (1990) Hydrogen Effects in Material Behaviour (eds N.R. Moody and A.W. Thompson) TMS, Warrendale, PA, p. 261. Hirabayashi, M., Yamaguchi, S., Asano, H. and Hiraga, K. (1974) Proc. Int. Conf. on Order–Disorder Transformations in Alloys (ed. H. Warlimont) p. 266. Holmberg, B. (1962) Acta Chem. Scand., 16, 1245. Hong S. and Fu C.L. (2002) Phys. Rev.(B), 66, 99. Jack, K.H. (1948) Proc. Roy. Soc., A195, 34. Jostsons, A. and McDougall (1970) Phys. Status Solidi, 29, 873. Khachaturyan, A.G. (1978) Prog. Mater. Sci. 22, 1. Khachaturyan, A.G. (1983) Theory of Structural Transformations in Solids, Wiley, New York. Lieberman, D.S., Read, T.A. and Wechsler, M.S. (1957) J. Appl. Phys., 28, 532. Leitch, B.W. and Puls, M.P. (1992) Metall. Trans., 23A, 797. Massalski, T.B. (1992) Binary Alloys Phase Diagram, 2nd edition, ASM International Materials Park, OH, pp. 2066–2067, 1652–1655, 1792–1793, 2078–2079, 1659–1660, 1799–1800. Narang, P.P., Paul, G.L. and Taylor, K.N.R. (1977) J. Less-Common Metals, 56, 125. Norwood, D.O. and Gilbert, R.W. (1978) J. Nucl. Mater., 78, 112. Norwood, D.O. and Kosasih, U. (1983) Int. Metals Rev., 28(2), 92. Pan, Z.L., Ritchie, I.G. and Puls, M.P. (1996) J. Nucl. Mater., 228(2), 227. Park, J.M. and Lee, J.Y. (1990) J. Less-Common Metals, 160, 259. Puls, M.P. (1990) Metall. Trans. A, 21A, 2905. Rhodes, C.G. and Williams, J.C. (1975) Metall. Trans. 6A, 1071, 2103. Shaltiel, D., Jacon, I. and Davidov, D. (1977) J. Less-Common Metals, 53, 117. Sidhu, S.S., Heaton, Le Roy, Campos, F.P. and Zauberis, D.D. (1963) Am. Chem. Soc., 39 Singh, R.N., Kumar, N., Kishore, R., Roychaudhary, S. and Sinha, T.K. (2002a) J. Nucl. Mater., 304, 189. Singh, R.N., Kishore, R., Sinha, T.K. and Kashyap, B.P. (2002b) J. Nucl. Mater., 301, 153. Singh, R.N., Kishore, R., Sinha, T.K., Banerjee, S. and Kashyap, B.P. (2003) Mater. Sci. Eng. A, 339, 17. Singh, R.N. Kishore, R. Singh, S.S., Sinha, T.K. and Kashyap, B.P. (2004) J. Nucl. Mater., 325, 26.


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Singh, R.N., Mukherjee, S., Gupta, A. and Banerjee, S. (2005) J. Alloys Compd., 389, 102. Singh, R.N., Lala Mikin, R., Dey, G.K., Sah, D.N., Batra, I.S. and Stahle, P. (2006) J. Nucl. Mater., 359, 208. Sinha, V.K., Shetty, M.N. and Singh, K.P. (1970) Trans. Faraday Soc., 66, 1981. Sinha, V.K., Pourarian, F. and Wallace, W.E. (1982) J. Less-Common Metals, 87, 283. Sinha, V.K., Yu, G.Y. and Wallace, W.E. (1985) J. Less-Common Metals, 106, 67. Srivastava, D. (1996) Ph.D Thesis, Indian Institute of Science, Bangalore, India. Srivastava, D., Neogy, S., Dey, G.K., Banerjee, S. and Ranganathan, S. (2005) Mater. Sci. Eng. A, 397, 138. Sundararaman, D., Terrance, A.L.E., Seetharaman, V. and Raghunathan, V.S. (1980) Titanium 80, Science and Technology, Proc. Fourth Int. Conf. on Titanium, Vol. 2, The Met. Society of AIME, New York, p. 1521. Sundararaman, D., Terrance, A.L.E., Van Tendeloo, G. and Van Landuyt, J. (1983), J. Phys. Status Solidi (A), 76, K-109. Sundararaman, D., Raju, S. and Raghunathan, V.S. (1989) J. Phys. Chem. Solids, 50, 1101. Une, K., Nogita, K., Ishimoto, S. and Ogata, K. (2004) J. Nucl. Sci. Technol. 41, 731. Taizhong, H., Zhu, W., Jinzhou, C., Xuebin, Y., Baojia X., Naixin, X. (2004) Mat. Sci. Eng. (A), 385, 17. Unnikrishnan, M., Menon, E.S.K. and Banerjee, S. (1978) J. Mater. Sci., 13, 1401. Wayman, C.M. (1964) Crystallography of Martensitic Transformation, Mcmillan, New York. Weatherly, G.C. (1981) Acta Metall., 29, 501. Yamaguchi, S., Hiraga, K. and Hirabayashi, M. (1970) J. Phys. Soc. Japan 28, 1014. Zener, C. (1946) Trans. AIME, 167, 550. Zuzek, E. and Abriata, A. (2000) Phase Diagrams of Binary Hydrogen Alloys, Monograph Series on Alloy Phase Diagrams, Vol. 13, ASM International, Materials Park, OH.


Chapter 9

Epilogue


Chapter 9

Epilogue

The preceding eight chapters have illustrated the wide variety of phase transformations encountered in Ti- and Zr-based systems and have demonstrated that the whole subject of phase transformations in condensed matter can be studied using examples taken from these systems. It has also been shown that the various concepts and formalisms introduced in the study of phase transformations in alloys can be applied to intermetallics and ceramics as well. In this concluding chapter, let us examine some general trends in the phase transformations in Ti- and Zr-based alloys. As has been discussed in earlier chapters, solid–solid phase transformations in these systems are essentially governed by the competition between the different allotropes, , and that are structurally related through unique lattice correspondences, the Burgers relation for / and that involving the collapse of a set of adjacent {222} planes for /. In fact, the structural relationships between these phases are so strong that the same relationships remain valid for both displacive and diffusional transformations. An inspection of phase diagrams of binary and ternary alloys reveals that in a fairly large number of Ti, Zr–transition metal systems, the liquidus temperature drops down considerably with alloy additions. This enhanced stability of the liquid phase makes it possible to amorphize these alloys under non-equilibrium processing conditions. The chemical short-range order present in some of these liquid phases which are stabilized to sufficiently low melting temperatures tends to form clusters with icosahedral symmetry. Such tendencies are reflected in the formation of quasicrystalline phases on crystallization of some of the amorphous alloys in these systems. Another strong tendency which is responsible for several phase reactions in these systems is the clustering tendency in the -phase. Spinodal decomposition, phase separation, monotectoid decomposition, metastable phase reactions during tempering of some martensites are all consequences of the clustering tendency in the -phase in a number of alloy systems. Tendencies for chemical ordering in the -, - and -phases, present in several binary and multicomponent systems, are responsible for the formation of ordered derivations of these phases. Important examples of such chemical ordering are hcp → D019 , bcc → B2 and → B82 . Symmetry relationships between the disordered parent structures and their ordered derivatives play a dominant role in dictating the path of the transformation sequence in several alloy systems. 785

786


The hcp structure is a special case √ of the orthorhombic structure in which the ratio of lattice parameters b/a = 3. With the introduction of some alloying elements beyond certain limits, the hcp structure is distorted to an orthorhombic √ symmetry (b/a = 3). This tendency of orthorhombic distortion is also present in some ordered intermetallics having the D019 structure which undergoes a transition to the O-phase (orthorhombic) following the same lattice correspondence as that prevailing between hcp and orthorhombic structures. We will briefly discuss these tendencies which determine the phase transformation sequences in Ti- and Zr-based systems. Pressure–temperature phase diagrams of single component systems comprising Group IVA metals, Ti, Zr and Hf show the three phases, hcp, bcc and (hexagonal), in the solid state. The most interesting aspects of these phases are their unique lattice correspondences and the relationship of their stability regimes with their electronic density of state. Introduction of alloying elements and/or change in external variables such as pressure and temperature can induce a change in the electronic structure which in turn is responsible for bringing about the phase transition. Out of the three allotropes, the -phase with the bcc structure is associated with the highest symmetry. This is also the first phase to appear from the liquid phase on cooling. Let us, therefore, consider the -phase as the reference state and examine how this bcc structure transforms into the hcp - and the hexagonal -structures. As has been elaborated in Chapters 1 and 4, the -phase experiences a tendency to undergo a transition into the -phase as the / transformation temperature is approached. The lattice correspondence between and is such that a distortion of a {110} plane can introduce a sixfold symmetry, and a rather small distortion along the direction can establish the right c/a ratio of the product hcp structure. In addition to this lattice strain, an atomic shuffle is necessary for bringing the atoms in every alternate basal planes in the right position. The -phase in some alloys cannot maintain the hexagonal symmetry and distorts into an orthorhombic structure. In addition to the tendency for the bcc to orthohexagonal transition, the -phase experiences another type of lattice instability which can be described as a longitudinal displacement wave of wave vector k = 2/3 . The introduction of this periodic displacement in the -lattice generates the -structure as has been discussed in Chapters 1 and 6. Variations in external conditions such as pressure and temperature and chemical composition shift the relative stabilities of these phases and the corresponding transformations can be induced. These two transformation tendencies lead to either an orthohexagonal structure or to the -structure. These tendencies are present not only in dilute alloys but also in several intermetallic compounds as has been elaborated in Chapters 4–6.

Epilogue

787

The -phase in several Ti and Zr alloys also exhibits a tendency for phase separation as reflected in the presence of a miscibility gap in the -phase field and a monotectoid reaction of the type 1 → + 2 . This phase separation or clustering tendency is instrumental in giving rise to a number of metastable steps in the phase transformation sequence of both - and -alloys. Chemical ordering of -, - and -phases can produce a host of ordered intermetallic structures in binary and ternary alloys of Ti and Zr. A comparison between several equilibrium and metastable intermetallic structures in these systems vis-àvis the -, - and -structures reveals that the former can be construed as ordered derivatives of the latter. The mechanism of formation of these intermetallic phases from the parent -phase, therefore, involves displacive atom movements (either from bcc to orthohexagonal or from bcc to ) in conjunction with replacive (chemical) ordering. The path associated with such transformations can be determined considering the symmetry changes associated with the displacive and replacive ordering processes. In spite of the diversity of solid-state phase transformation processes in Ti- and Zr-based alloys, an understanding of the mechanism involved can often be gained by taking into the account the fact that the structures are invariably closely related and that the transformations are often guided by the following general tendencies: (1) bcc → orthohexagonal lattice distortions, (2) bcc → transformation by the shuffle dominated lattice collapse process, (3) phase separation tendency of the -phase in several systems containing -stabilizing elements and (4) chemical ordering tendencies of the -, - and -phases leading to the formation of their respective superlattice structures. Coming to the issue of the formation of amorphous and quasicrystalline structures, the stabilization of the liquid phase by alloying Ti and Zr with various transition metals is reflected in the plunging of the liquidus temperature in the respective phase diagrams. In case the formation of the equilibrium intermetallic phases is suppressed by a non-equilibrium processing technique at a temperature below the glass transition temperature, the amorphous phase becomes the next best alternative for the system to adopt. The presence of the chemical short-range order in the liquid phase in compositions close to those of relevant intermetallic compounds is a prerequisite for bringing about vitrification in these systems. The nature of clusters is undoubtedly very important in promoting the stability of the amorphous phase as is seen from the formation of bulk glasses in a host of ternary and quaternary Zr–Al–transition metal alloys. These clusters, some of which have icosahedral or decagonal symmetries, have a tendency to retard the

788


kinetics of crystallization, thereby making the corresponding alloys amenable for glass formation even at a relatively slow cooling rate. In order to identify different general tendencies of phase transformations, some recent experimental techniques have been found to be extremely convenient. To illustrate this point let us cite the examples of studies carried out on miniature compositionally graded samples (Banerjee et al. 2003). Phase transformation studies on a given binary or multicomponent alloy system invariably necessitate the preparation of a series of alloys of different compositions and thermal analysis, phase analysis and microstructural characterization of different microconstituents after suitable heat treatments. This has been the general practice till recently; now it is possible to produce compositionally graded samples with desired gradients by laser melting of controlled powder feeds emanating from multiple nozzles. Figure 9.1 illustrates the operating principle of such a laserprocessing unit. In this process, either prealloyed powder or elemental powders can be taken as the powder feed. When powders of different compositions are fed from separate feeders with a precise control of feed rates at a small size molten puddle, the possibility of making samples with graded composition opens up. The process control computer continuously controls the position (x y z) of the hot spot (molten puddle) with respect to the component/sample being processed and simultaneously controls the feed rates of powders from different nozzles. Such a manufacturing process (which is essentially an extension of rapid prototyping process) has the potential not only of producing components of intricate shapes, but also of preparing samples of desired compositional gradients. Glove box CO2 laser system Powder feed system

Stage control

Figure 9.1. Experimental set-up for production of compositionally graded samples.

Epilogue

789

Suitable slices of one such graded sample can be prepared and given appropriate heat treatments for inducing different phase transformations. Microscopic examination of graded samples by a scanning electron microscope (SEM) attached with microanalysis and orientation imaging facilities can yield morphological, crystallographic and compositional information on the microconstituents at a level of about 0.1 m resolution. Such studies can be complemented by TEM studies by picking up samples from related areas by focussed ion beam sectionizing (FIBS) technique. Nanoindentation experiments on small sample areas can further supplement the investigation by providing mechanical properties data from each of the microconstituents. Recent work reported on Ti–V and Ti–Mo alloys have shown how these techniques can be gainfully employed for phase transformation studies on a system from a single compositionally graded sample (Banerjee et al. 2002, 2003). The composition profile of a graded Ti–V sample of 25 mm length produced by laser melting of prealloyed powders of different compositions is shown in Figure 9.2. Several longitudinal slices of such a sample after subjecting to different heat treatments such as (a) as cast, (b) -quenched and (c) -solutionized followed by air cooled show a variety of microstructures as described in the following. The -quenched sample shows the presence of lath martensite in regions corresponding to a V level of 0–3 at.%, plate martensite in the region of 3–11 at.% V

Ti–25 at.% V

Ti

∼ 25 mm 25

#14 #13

Composition (at.% V)

20 #12 15 #10

#11

#9 10

#8 #7 #6

5 #4

#3 0

#5

#1 #2 0

5

10

15

20

25

Distance (mm)

Figure 9.2. Variation of composition of a Ti–V compositionally graded sample.

30

790


and near complete -stabilization in regions containing more than 12% V. TEM investigations have revealed that a distribution of fine -plates in the composition range of 12–15% V. Microstructures of the as-cast sample match closely with those corresponding to -solutionizing and air cooling. This points out that the laser-processing technique on a sample size, like the present one, essentially produces a normalized structure. Microstructures of the compositionally graded samples (both as-cast and -solutionized followed by air cooled) show a gradual change in the morphology of Widmanstatten -plates and monoatomic increase in the volume fraction of the retained -phase with increasing V content (Figure 9.3). While the -plates are long and slender in alloys containing up to about 5 at.%, they are short and stout in alloys more enriched with V. Though the as-cast condition is not expected to yield equilibrium microstructure, the volume fractions of the - and the -phases and their respective compositions are seen to approach equilibrium values particularly in samples lean in V. With an increase in the V level, as the / + transus temperature comes down, the departure from the equilibrium values of volume fraction and composition is larger, suggesting a more incomplete partitioning of alloying elements in the - and the -phases. The reduction in the length of the plates is also consistent with a slower growth kinetics of the plates and an increased gap in the composition of and . In regions corresponding to V level of about 5–8 at.%, the presence of a bimodal size distribution of -plates can be attributed to a two-stage decomposition process – primary plate forming when the temperature of deposited alloy drops below the / + transus and the secondary plates subsequently appearing in the retained -phase during later heating cycles. Crystallographic information from these samples at a mesoscopic level (typically at a magnification of 2000×) obtained from orientation imaging microscopy (01 m) has yielded an orientation distribution map for the matrix and the different variants of the -phase (Figure 9.4). As per the Burgers relationship, six equivalent variants of planar matching exist, where the basal (0001) plane is parallel to one of the six variants of the 110 plane of the -phase. Again for a given variant of planar matching (say (0001) (110) ), there are two distinct options of directional matching, namely [1120] [111] and [1210] [111] . OIM has identified all the 12 orientation variants and provided quantitatively the area fractions occupied by each of them. The orientation distribution information presented in pole figures (Figure 9.4) corresponding to {110} , {111} , (0001) and 1120 poles have oneto-one correspondence with the 110 poles and the 111 poles matched with the 1120 poles. Observations like this not only validate the Burgers relationship, but also show special disposition of orientation variants in specific locations. It has been shown recently (Bhattacharyya et al. 2003) that -phase forming at grain boundaries (described in Chapter 7 as “Grain Boundary Allotriomorphs”)

Epilogue

791

10 μm

Ti–1.8 at.% V

Ti–5 at.% V

Ti–3 at.% V

Ti–6.8 at.% V

10 μm

Ti–8 at.% V

Ti–10 at.% V

Figure 9.3. Laser-processed (as-cast) sample with composition gradient shows variation in the aspect ratio of -plates and in the volume fraction of the -matrix (lighter shade) with increasing V contents.

often correspond to a stack of two alternate orientation variants which have the same planar matching (say (0001) (110) ) with directional matching alternating between [1120] [111] and [1210] [111] . According to the nomenclature introduced in describing martensite variants (Chapter 4) these are designated as ( − +) and ( + +), respectively.

792


Figure 9.4. Distribution of different orientation variants of -plates in -matrix as revealed from orientation imaging microscopy.

Microstructural examinations of a second slice of the same compositionally graded Ti–V bar – which has been given the -quenching treatment – reveal the presence of lath martensite in the region corresponding to 0–3% V, plate martensite in the region having 3–11% V and nearly complete -phase stabilization in the region of V > 12%. Microstructural observations on a finer scale from selected regions have become feasible due to the availability of FIBS technique. TEM examination of samples picked up from regions of interest reveals the following:

Epilogue

793

g = 1012

2 μm

500 nm

Figure 9.5. Parallel stacking of -laths with small angle boundaries laths.

(1) Regions of the laser-processed sample in the composition of 1–2% V show Widmanstatten -laths, contiguously stacked with small angle boundaries between the adjacent laths. In addition to the interfacial dislocations, an array of discrete fine -plates are seen (Figure 9.5) along these interlath boundaries. These fine plates again exhibit an internal substructure akin to martensitic substructure (Figure 9.6). Based on this observation, Banerjee et al. (2003) have proposed that the interlath plates indeed form by a martensitic process. This is possible because under the non-equilibrium cooling condition prevailing during laser processing, partitioning of V in the - and the -phases is not complete. Under such a situation, the metastable + / transus can intersect the Ms line making the martensitic transformation possible within the -layer at the interlath boundary. (2) In regions having average V content above 5%, primary and secondary -plates are seen to be distributed in the -matrix (Figure 9.7) which exhibits a mottled structure. Diffraction patterns corresponding to the -matrix clearly indicate the presence of both - and -reflections. For the region corresponding to an average composition of 5% V, the -matrix (containing 15% V) shows a distribution of fine - and -particles. With further V enrichment of the -matrix (1˜ 7.5% V), only -particles are seen to be present. This is consistent with the general trend of enhancement of the stability of in comparison to that of as the -phase is enriched with V. (3) In regions containing average V content above 20%, the -phase is retained. However, the -phase with 20% V has been found to be amenable to stressinduced → transformation as revealed from nanoindentation experiments. Experimental results on the microstructure of laser-processed compositionally graded sample of about 1 in. length are presented here to illustrate that some of the major trends of phase transformations of titanium (and also zirconium) alloys can all be identified from such a small sample. Some of the recent experimental

794


g = 0002

g = 1010

100 nm

g = 1011

g = 1011

200 nm

Figure 9.6. Vanadium-enriched interlath regions exhibiting martensite-like substructure.

techniques have made it possible to study a whole range of phenomena in a single sample. Metallurgists for many years have been using Jominy test bars for achieving graded quenching rates in a single sample. The present trend of combinatorial materials science can explore, in a similar manner, a wide range of phase transformations in a single compositionally graded sample. Such experimental techniques will no doubt complement the recent theoretical development of first principle predictions of phase stability of alloys. In this concluding chapter, it is worthwhile to draw a comparison between the observed crystallographic features of diffusional, martensitic and mixed-mode transformations encountered in titanium and zirconium alloy. As mentioned earlier, an approximate − Burgers orientation relationship is a dominant feature in all types of transformations discussed, namely, martensitic transformation, Widmanstatten -precipitation, -hydride precipitation and active eutectoid decomposition. This statement, however, is being made by not distinguishing the

Epilogue

Figure 9.7. -plates in V-stabilized -matrix containing dispersion of -particles.

795

796


Burgers and the Potter relations, which differ by a rotation of 1 5 . The range of transformations discussed involves shear, diffusional and mixed-mode mechanisms, and it is attractive to examine the applicability of the invariant line strain (ILS) and the invariant plane strain (IPS) criteria in determining the habit planes of advancing transformation fronts associated with these transformations. The macroscopic habit plane for the martensitic plates and laths can be precisely predicted on the basis of the IPS criterion. The interfaces of a martensite lath consist ¯ 1123 ¯ of arrays of c + a dislocations, suggesting the operation of the 1011 slip mode as the lattice invariant shear (LIS). Since the LIS is accomplished by the passage of these dislocations, it is necessary for the line vectors of these dislocations to be along the invariant (IL) vector. The lath boundary structure, however, is more complex, as in the lath martensites, the parent -phase is fully consumed and adjacent plates come in contact, resulting in a superimposition of their interface structures. In twinned martensitic plates for both ( − +) and ( + +) solutions, for the two twin-related variants to remain in coherence with the parent , it is geometrically necessary for them to meet along the IL vector. The same argument is also valid for a group of three martensite crystals that form the indentation morphology. These self-accommodating groups of crystals meet along a line that is the IL. The line of intersection of habit plane segments corresponding to two adjacent twin-related martensite variants within a twinned plate following the ( − +) solution of the Bowles–Mackenzie analysis matches the ILS direction. Martensite plates that obey the ( + +) solution contain a stack of thin twins, which again intersect the habit planes along the ILS direction. The geometries of the habit planes in these cases are schematically illustrated in Figure 9.8(a) and (b). The habit planes of Widmanstatten laths/plates have also been found to be irrational. However, habit plane predictions based on the IPS criterion do not match those experimentally observed. Two types of Widmanstatten products, namely, single crystal laths and internally twinned plates, have been encountered. The lath interface exhibits an array of dislocations with c + a Burgers vectors, where dislocation line vectors are parallel to the ILS direction. In fact, the growth direction of the laths is marked by the line vector of these interfacial dislocations, which lie along all four surfaces parallel to the long axis of the laths. It is also seen that the ¯ habit plane poles are located in the vicinity of the pole, where [112] and 1010 come in coincidence, consequent to the operation of the Burgers relation. This observation is consistent with that of Furuhara and Aaronson (1991) who reported that the habit of (hcp) plates in the matrix of (bcc) Ti–Cr alloys lies close to ¯ directions. 130 and 131 poles, which are located near the [112] 1010 They also invoked a periodic occurrence of structural ledges, which accounts for

Epilogue IL

IL

Habit (301)β-(301)β

)β

33

(4

)β 33 (4

797

(1011)α

(a)

(d)

α Precipitate in β

1

(21

IL

Average habit (301)β-(311)β

(1011)α//(110)β

(1011)α

β )β

2) β

(b)

(11

Martensite Class A (α−ω+) thick 1011 twins IL

(e)

α Precipitate in β

Martensite Class (α+ω+) thin 1011 twins

IL

IL

Transformation front

α

α

α

β′

γ Hydrie in β

(101

1)α/

α

/(11

)γ

(f)

(111

(c)

0)β

Habit (301)β-(311)β

Active Eutectoid Decomposition β→α+β′

Figure 9.8. A schematic illustration of habit planes/transformation fronts in various types of diffusional, martensitic and mixed-mode processes. The role of the IL vector in all processes is revealed.

¯ the rotation of the average habit interface from the terrace plane (112) 1010 on which patches of good atomic fit across the / interface exist. The criterion of the selection of the habit plane in a diffusional transformation has been discussed in Banerjee et al. (1997) and references therein. There is a general agreement that one of the vectors that defines the habit plane is the ILS direction. The second vector fulfils one of the following conditions: • • • •

a vector that remains unrotated, a direction of low strain, a direction of easy slip in the parent or product phase, a direction favoured due to the elastic anisotropy of the matrix–precipitate assembly and • a direction along which structural ledges are aligned.

798


In a recent article, Zhang and Purdy (1993a,b) have shown that using an extension of the concept of the O-lattice to a system containing invariant lines, a complete Burgers vector balance can be accomplished at the / interface of the Zr–2.5Nb alloy by a single set of dislocations with the Burgers vector [010] and dislocation spacing of 10 nm. This prediction remains valid for Zr–20Nb also, as the lattice parameters of the Zr–20Nb alloy are not significantly different than those for the Zr–2.5Nb alloy. The observed interfacial structure (a single set of parallel dislocations along the ILS direction) (see Figure 7.91) and the experimentally determined habit plane indices are consistent with the predictions of Zhang and Purdy (1993a,b) who have shown that the habit plane is given by an O-lattice plane associated with a plane of the reference lattice, which contains the Burgers vector of the dislocations. The habit plane envisaged in this case differs from that predicted from the atomistic model (see Section 7.5.4) proposed in a similar system of Ti–Cr alloys, though the indices of the macroscopic habit plane and the orientation relation are quite similar in the two cases. Matching of the atomic planes across the / boundary in Ti–Cr alloys has been achieved by the introduction of biatomic structural ledges with Burgers vector 16 a and of misfit compensating c-type ledges. It appears that in the case of the / interface in Zr–Nb alloys, the misfit along the c and a components is compensated by a single set of c + a dislocations lying along the ILS direction (Figure 9.8(c) and 7.91). Examples of internally twinned Widmanstatten -plates have been reported in ¯ twinning and a habit plane pole Banerjee et al. (1997). These plates exhibit 1011 lying between 130 and 131 . This habit does not match that predicted from the IPS consideration. A detailed analysis of the macroscopic facets that constitute the average habit plane could not be performed with sufficient accuracy. However, ¯ planes. It is, therefore, the major facet appeared to be close to the [112] [1010] quite attractive to consider that the macroscopic habit plane is made up of steps of ¯ planes on which good atomic fit can be established. The direction [112] [1010] along which two adjacent twin variants meet on the habit plane has been found to be the ILS direction (Figure 9.8(d)). The crystallography of precipitation of the -hydride plates in either the - or matrix obeys the IPS criterion. As mentioned earlier, this transformation involves a shear transformation of the or zirconium lattice, with the accompanying process of hydrogen partitioning. For such a transformation, the phenomenological theory of martensite crystallography provides an accurate prediction of the habit plane. The habit plane retains the direction of the IL, which is defined by the line of intersection of the habit plane and the twin plane (Figure 9.8(e)). Active eutectoid decomposition of the parent -phase occurs through a cooperative growth of the - and metastable -phases. Based on crystallographic

Epilogue

799

observations, it can be inferred that the Burgers relation exists between and , while the / relationship is based on the parallelism between their respective ¯ ) cube axes. The line along which the / interface (which is parallel to 1011 intersects the transformation front is the IL vector (Figure 9.8(f)). Crystallographic descriptions of different types of transformation products, as schematically illustrated in Figure 9.8, bring out the importance of the IL vector in defining the habit plane (or the transformation front). The tendency of maintaining at least one undistorted vector on the advancing transformation is not unexpected. It is increasingly being realized that either partial or full coherency is maintained in a great majority of solid-state phase transformations, both martensitic and diffusional. This point has been amply demonstrated in the case of transformations in titanium- and zirconium-based alloys. The operation of Burgers or near Burgers orientation relations has an overwhelming influence on all types of transformations in titanium- and zirconium-based alloy systems. The diffusional transformations are, however, distinguishable from displacive transformations from the nature of the habit plane. While the habit plane fulfilling the IPS criterion is a necessity in displacive or diffusional-displace (e.g. formation of -hydride) processes, the habit planes in pure diffusional transformations are determined by the strong tendency of maintaining regions of good fit between the parent and the product phases. Since in the latter case interfaces often deviate from being planar, a rotation of the boundary is achieved by rotating the interface around the invariant line vector. A typical -lath formed through a diffusional process possesses an interface characterized by an array of c + a dislocations, which are all aligned along the IL vector. Even for the active eutectoid decomposition, which involves partitioning of substitutional alloying elements, the two product phases appear to grow in such a manner that the crystallographic registry of the parent is maintained simultaneously with two product crystals. The velocity of interface motion is, therefore, limited by the requirements of chemical diffusion for attaining the appropriate extent of alloy partitioning and for the ordering processes within one of the product phases. In the preface of the book, we have highlighted the diversity of phase transformations in titanium- and zirconium-based alloys and subsequently have justified that the whole subject of phase transformations can be covered using examples taken from these alloys. Finally, we are only emphasizing that this wide range of transformations are driven by only a few trends of structural and chemical instabilities. It is also noted that different transformation mechanisms leave their imprints on the crystallographic and morphological features of the transformation products and it is through these experimental observables one deciphers the relevant operating mechanisms.

800


REFERENCES Banerjee, S., Dey, G.K., Srivastava, D. and Ranganathan, S. (1997) Metall. Trans., 28A, 2201. Banerjee, R., Collins, P.C., and Fraser, H.L. (2002) Metall. Mater. Trans. A, 33, 2129. Banerjee, R., Collins, P.C., Bhattacharayya, D., Banerjee, S. and Fraser, H.L. (2003) Acta Mater., 51, 3277. Bhattacharyya, D., Viswanathan, G.B., Denkuberger, R., Furrer, D. and Fraser, H.L. (2003) Acta Mater., 51, 4679. Furuhara, T. and Aaronson, H.I. (1991) Acta Metall. Mater., 39, 2857. Zhang, W.Z. and Purdy, G.R. (1993a) Acta Metall. Mater., 41, 543. Zhang, W.Z. and Purdy, G.R. (1993b), Philos. Mag., 68, 291.

Index -TiCr2 -TiCr2

alloys 21, 22, 284, 687 alloys 21, 284, 504–506, 539, 567, 689–690, 701 -hydride 120, 721, 722, 728, 729, 730, 733, 736, 737, 738, 794, 798 -hydride (fct) phase 721 -hydride 722, 723 -hydride 722, 723, 728, 739 -phase immiscibility 606–609 -phase 5, 10, 21, 26, 49, 189, 217, 223, 447, 529, 656, 683 -phase 5, 9, 13, 21, 22, 25, 43, 48, 50, 52–53, 134, 145, 283, 310, 384, 421, 447, 475–484, 486, 488, 512, 523, 559, 599, 609–615, 632, 655, 688, 739, 753, 787 -phase 6, 7, 9, 10, 12, 13, 26, 35, 36, 43, 44, 50, 52, 251, 284, 490, 492–494, 504, 512, 516, 529, 536, 544 -precipitation 492, 536, 537, 539–540 -stabilizers 21, 22, 154 -stabilizers 21, 25, 53, 174, 325, 383, 417 -stabilizing elements 21–23, 148, 175, 251, 281, 303, 310, 417, 475, 516, 587, 609 -transformed 688, 689, 693, 694, 698, 699, 700 -transition 9, 12, 106, 419, 474, 481–484, 491–495, 520 + alloys 21, 22, 284, 327, 609, 632, 687–689, 696, 729 – lattice correspondence 485 – lattice correspondence 488 –-isomorphous systems 27 / interface 559, 646, 647, 650, 657, 737, 738, 799 2 -phase 382, 383, 418, 420, 421, 423, 434, 443, 445, 447, 448, 451, 453, 458, 662–670 -eutectoid systems 27, 622 -isomorphous systems 27, 29, 30, 606, 609, 620 -TiCr2 33, 62

33, 62 33, 62, 622

A15 (cP8, Cr3 Si type) 54–62 ab initio phase diagram calculations 16 Accommodation strain energy 719, 725, 726 Accommodation stress 264 “Active” eutectoid decomposition 560 Affine transformation 103, 267 Aged 45, 50, 475, 476, 492, 538, 547 Ageing 44, 50, 53, 421, 435, 448, 453, 475, 529, 533, 539, 546, 603, 658, 706, 728, 772 Allotrope 5, 286 Amorphous alloys 162, 163, 165, 176–179, 183, 204, 205, 207, 209, 210, 250 Amorphous phase 160, 161, 164, 165, 175, 176, 178, 182, 183, 185–187, 191, 192, 203, 207, 208, 210, 217, 220–229, 232, 233 Anomalous diffusion behaviour 13, 568 Antiferroelectric 109 Antiferromagnetic 109 Applied stress 263–265, 324, 331, 340, 350, 354, 355, 370, 725, 727, 747 Athermal 43, 44, 49–52, 101, 105, 107, 109, 110, 264, 265, 277, 278, 280–283, 338, 339, 362, 474–477, 486, 490–492, 494, 495, 500, 501, 508–510, 512, 517, 529, 539, 540, 546, 560 Athermal → transition 50, 475, 486, 491, 501, 512 Athermal martensitic 43, 278 Athermal nucleation 110 Atomic layer stacking 55 Atomic shuffles 111, 316, 320, 504, 505 Atomic site correspondence 632, 650 Austenite 47, 121, 153, 260–262, 267, 269, 270, 279, 324, 325, 354, 355, 358, 359, 371, 673

801

802 Autocatalytic 305, 322 Avrami exponent 193, 197, 202, 204, 211 B2 (cP2, CsCl type) 54, 60, 61 B82 141, 156, 210, 441, 518, 519, 521, 525, 527, 528, 529, 530, 533, 535, 785 Bain distortion 269, 272, 292–296, 298 Bain strain 45, 269, 273, 274, 276, 293, 294, 314, 316, 321, 431, 432, 504, 643 Bainite-like transformation 742 Bainitic transformation 121, 122, 737 Ball milling 226, 228, 229 Basket weave morphology 421, 637, 656 BaTiO3 111, 113, 114, 260 BCC special points 404 Bf (oC8, CrB type) 54, 60 Black plates 638, 648 Blister formation threshold 744, 751 Boiling water reactors 771 Bridge 351 Bulk Metallic Glasses 157, 158, 171, 205 Burgers 45, 46, 276, 289, 293, 294, 302–304, 310, 312, 314–316, 332, 334, 336, 339, 420, 421, 424, 538, 539, 547, 558, 606, 613, 614, 618, 633, 635, 642–644, 646, 648–651, 655, 679, 730, 736 Burgers correspondence 294, 421, 679, 730, 736 Burst 278, 362, 547, 550, 743 Burst-like behaviour 362 C14 (hP12, MgZn2 type) 54, 59, 61 C15 (cF24, Cu2 Mg type) 54, 59, 61 C16 (tI12, CuAl2 type) 60 C32 (hP3, AlB2 type) 60 Cellular precipitates 635 Ceramics 55, 62, 63, 90, 92, 106, 107, 261, 369, 371, 372, 558, 559 Chemical bonding 12 Chemical diffusion 555, 570, 578, 799 Chemical free energy 202, 238, 261, 263, 264, 278, 287, 324, 371, 456, 580, 592, 593, 612, 618, 675, 725, 726 Chemical potential 98, 129, 130, 132, 152, 285, 395, 530, 571, 577, 589

Index Chemical spinodal 558, 594, 601, 602, 621, 622 Classification 21, 27, 28, 57, 90–93, 101, 105–107, 111, 115, 283, 284, 540, 587, 633, 722 Clausius–Clapeyron 8, 359 Cluster approximation 394, 395, 396, 398, 399, 412 Cluster variation method 392 Coherency 65, 99, 240, 264, 265, 350, 370, 504, 593, 606, 614, 618, 633, 650, 662, 677 Coherent phase transformations 386 Coherent spinodal 99, 525, 558, 594, 601, 618, 620, 621, 622 Cold rolling 702 Composition-invariant transformations 91, 110, 435 Composition modulation 525, 593, 594, 616, 618 Composition-induced destabilization of crystalline phases 220 Compressive 290, 702, 769, 770 Concentration modulation 99, 381, 438, 557, 592, 602, 603, 622 Concentration wave 99, 117, 120, 183, 379, 390, 391, 392, 393, 406, 416, 437, 438, 459, 460, 524, 525, 527, 528, 606 Concomitant clustering and ordering 407 Configurational energy 393, 395, 776 Constitutional solution treatment 753 Continuous or homogeneous transitions 97 Cooling rate 48, 128, 141, 142, 144–151, 153, 158–160, 165, 171–173, 175, 180, 182, 205, 281, 383, 420, 421, 433, 434, 446, 448, 450, 452, 626, 627, 629, 655, 656, 708, 725, 728, 739, 753 Coordination factor 17 Coordination numbers 59, 775 Correlation functions 393, 395, 397, 398, 399 Correspondence matrices 487, 489, 730 Correspondence variant 294, 343, 348, 350, 665, 734 Corrosion 157, 586, 587, 659, 661, 702, 721, 741, 769, 771, 772

Index Coupled transformation 122 Cross slip 332, 333, 688, 698 Crystallization 92, 110, 111, 145, 157, 158, 160, 170, 172, 175–178, 181, 182, 184–189, 191–197, 200–205, 207, 209–211, 213, 220, 250 Crystallization kinetics 193, 200, 210, 211 Crystallographic texture 701, 702, 704, 706, 707, 743, 745 Crystallography 44, 50, 266, 269, 270, 282, 290, 292, 303, 320, 340, 342, 345, 366, 372, 418, 424, 474, 484, 534, 613–615, 648, 657, 662, 665, 720, 721, 728–730, 732, 735, 736 Cu-Zn-Al 362 d-band occupation 12 d-band 13, 18–20, 508, 516–518 D019 54, 55, 59, 61, 62, 227, 228, 229, 377, 382, 383, 407, 412, 413, 416, 417, 418, 420, 421, 423, 427, 428, 430, 431, 436, 437, 438, 439, 440, 441, 442, 444, 447, 451, 785, 786 D88 54, 60, 520, 521, 525, 527, 528, 529, 530, 535 Deformation twins 331, 334, 546 Degradation Processes 741 Degree of self-accommodation 308, 666 Delayed hydride cracking 742, 743, 744, 745, 746 Dendritic 137, 141, 145, 146, 150, 173, 175, 189, 203, 212, 690 Density functional theory 14, 15, 385 Density of states 18, 19, 384, 516 DHC 746, 747 DHC velocity 746 Diad 296, 346, 351 Differential scanning calorimetry 340 Diffraction effects 51 Diffuse Scattering 51, 112, 249, 471, 474, 495, 499, 533 Diffusion anisotropy difference 565 Diffusion bonding 555, 584, 585, 586 Diffusion mechanisms 157, 555, 560 Diffusion zone 559, 573, 574, 578, 579, 584, 585, 586, 587

803 Diffusional 43, 48, 50, 65, 101, 103–107, 109–111, 120–122, 129, 135, 148, 149, 185, 204, 266, 281, 293, 419, 453, 500, 504, 520, 558–560, 623, 625, 632, 642–645, 648, 650–653, 655, 657, 662, 683, 738, 739, 779 Diffusional transformation 103, 120, 558, 632, 643, 650, 785, 799 Dilation parameter 276 Discontinuous coarsening 450, 456, 457, 675 Discrete transformations 97, 109, 110 Dislocations 49, 121, 264, 276, 279, 280, 310, 312, 315, 316, 327, 329, 331–335, 337–339, 342, 387, 441, 451–455, 537–540, 546–548, 550, 562, 606, 626, 633, 645, 646, 648–652, 661, 677, 683, 684, 687–689, 692, 698, 710, 739, 753 Displacement ordering 98, 156, 419, 501–503, 509, 518, 550 Displacement 43, 101, 111, 114, 120, 431, 451, 506–508, 786 Displacement vector 316, 317, 319, 431, 432, 451, 454, 473 Displacement wave 43, 101, 120, 419, 427, 485, 486, 499, 500, 506, 510, 513, 527, 528, 531, 532, 533, 786 Displacive 43, 101, 109, 113, 120–122, 386, 389, 419, 425, 426, 429, 430, 492, 495, 518, 519, 525, 534 Displacive transformation 43, 100, 101, 103–107, 111, 120, 154, 261, 492, 518, 559, 650, 721 Distorted hexagonal 46, 281 Divacancies 561, 562 Domain structure 319, 419, 428, 767 Dynamic recovery (DRV) 683, 685, 686, 687, 691, 694 Dynamic recrystallization (DRX) 683, 687, 694, 698, 701, 706 Dynamic strain ageing 539, 540, 542, 546 Effective full power days (EFPD) 769 Elastic energy 263, 323 Electron concentration (e/a) factor 17 Electron diffraction 51, 52, 118, 174, 175, 341, 362, 495, 533, 603

804 Electron to atom (e/a) ratio 24 Electronegativity factor 17 Electronic structure 13–15, 24, 384, 385, 386, 393, 412, 413, 507, 510 Ellipsoidal inclusion 306 Embrittlement 181, 182, 471, 536, 537, 742, 746 Enantiotropic 92 Euler’s formula 300 Eutectic crystallization 111, 185, 187, 192, 194–197, 202–204 Eutectoid Decomposition 22, 27, 28, 65, 91, 111, 153, 186, 447, 448, 556, 560, 670, 671, 672, 673, 674, 675, 676, 681, 682, 722, 794, 798, 799 Exchange mechanisms 561 Exothermic 182, 184, 187, 189, 194, 207, 210, 340, 360, 361, 722 Extended defects 562 FCC special points 407 Fe24 Zr76 191 Fe30 Zr70 191 Fe40 Zr60 191 Fe–B 157, 162, 177, 187, 320, 356 Fe-Ni-Co-B 162 Ferroelectric 100, 106, 107, 111, 113, 114 Ferromagnetic 94, 107, 116, 162 Ferrous 47, 260, 263, 277, 279, 287, 320, 322, 326, 332 FeZr2 191 FeZr3 191 Fick’s laws 555, 562 First principles 11, 14, 17, 18, 117, 386, 393, 412, 503, 507, 510 First-order transformation 94, 97, 101, 105, 128, 262 First-order transitions 5, 93, 98, 107, 115 Flow localization 685, 691, 695, 701 Flow softening 691, 693, 696, 698, 701 Flow stresses 260, 328, 710 Fractal morphology 321, 637 Fractal 308, 313, 637 Fracture 64, 145, 226, 327, 352, 353, 370, 371, 537, 539, 547, 655, 690, 691, 721, 742, 747

Index Gas atomization 150 Gd–Co40–50 163 Gd–Fe32–50 163 Geometrically close packed (GCP) structures 57 Glass formation in diffusion couples 215, 220 Glass formation 134, 157, 158, 160, 161, 165, 168, 171, 205, 206, 213, 215, 220, 226 Glass transition 158, 161, 181–185, 193, 194, 200, 205, 209, 210, 218 Glass-forming abilities (GFAs) 157, 158, 160, 171 Glissile 105, 120, 121, 277, 278, 350, 651, 680 Gliding 332, 337, 504 Glissile interface 105, 120, 121, 278 Grain boundary allotriomorphs 633, 664, 674, 690 Grain-boundary diffusion 769 Ground states 377, 381, 397, 398, 403 Group number 18, 24, 26, 668, 669 Group/subgroup relations 5, 377, 424 Growth 49, 97, 98, 101, 105, 109, 110, 113, 116, 120, 122, 128, 129, 135–137, 139, 140, 144–146, 155, 158, 162, 164, 165, 172, 175, 177, 185, 188, 189, 191–194, 196, 197, 199, 200, 202–206, 208, 211, 212, 223, 225, 264, 265, 277–280, 308, 314, 315, 322, 355, 358, 362, 381, 421, 442, 451–453, 455, 457, 492, 494, 499, 504, 510,527, 539, 547, 559, 581, 582, 585, 597, 614, 623, 625, 626, 628–630, 634, 635, 637, 639, 641, 648, 650–653, 655, 657, 658, 661, 666, 668, 669–671, 673, 674, 681, 683, 698, 702, 707, 721, 726, 746, 747, 751, 769 Growth ledges 637, 651, 652 Habit plane 45, 109, 110, 270, 293, 303, 314, 348, 366, 368, 666, 732, 799 Habit plane variants 277, 313, 314, 348, 350, 352, 677, 679 Hafnium (Hf) 4 Hall–Petch constant 337, 338 Hall–Petch relation 337, 338 Hardening 326, 327, 329, 335, 336, 337, 339, 341, 536, 538, 604, 662, 683, 687 HCP special points 406

Index Heterogeneous 165, 169, 171, 175, 183, 213, 221, 222, 279, 381, 426, 451, 455, 533, 620, 625, 640, 661, 689 Heterogeneous nucleation 165, 169, 171, 175, 213, 222, 279, 455, 553, 620, 640, 661 Heterophase fluctuation 101, 279, 589 Hexagonal, H net 55 Homogeneous deformation 102, 103, 267, 268, 343, 345, 505, 643, 689, 735 Homogeneous nucleation 165, 170–173, 206, 278, 279, 527, 625 Homogenization treatment 758 Homophase fluctuation 589, 594 Hume-Rothery phases 384 Hydride blister 742,743, 744, 747, 749, 751 Hydride blister formation 742, 749 Hydride precipitation 717, 721, 726, 727, 736, 742, 747, 794 Hydrogen absorption 754, 755, 758, 760, 761 Hydrogen charging 225 Hydrogen ingress 738, 741, 742 Hydrogen migration 737, 743, 744, 746, 747 Hydrogen storage capacity 756, 758, 761, 762, 764 Hydrogen storage materials 55, 62, 754, 756, 758, 759, 760, 761, 764 Hydrostatic 215, 479, 494, 499, 508, 719, 720, 727, 744 Hysteresis 9, 10, 50, 353, 354, 356, 362, 475, 479–481, 726, 756, 758, 761, 764 Icosahedral phases 248–251 Incommensurate -structures 509, 515 Inhomogeneous shear 45, 47, 49, 275 Instability diagrams 410 Instability temperature 98, 107, 115, 528 Interaction energies 14, 380, 403, 427, 465, 727, 775 Interdiffusion 127, 128, 204, 224, 225, 555, 557, 559, 570, 572, 573, 574, 576, 577, 578, 585, 592, 604 Interface instability 146, 657 Interface phase 737, 738 Interfacial energy 206, 238, 263, 264, 280, 456, 458, 504, 580, 592, 604, 644, 657, 725 Interfacial equilibrium 129

805 Interfacial structure 555, 648, 650, 652, 798 Interlayers 225, 585, 586 Intermetallics 17, 54, 55, 57, 61, 90, 92, 106, 107, 111, 231, 261, 356, 382, 474, 558, 662, 669, 754 Intermetallic compounds 22, 153, 218, 225, 231, 414, 458, 559, 587, 657, 758, 787 Intermetallic Phases 26, 33–35, 38, 43, 53–55, 57, 59, 61, 62, 160, 162, 164, 175, 188, 284, 384, 414, 443, 518, 525, 560, 615, 638, 657, 658, 676, 754 Internal twinning 728, 729, 737 Internal twins 267, 276, 293, 304, 312, 316, 347, 635, 736 Interstitial ordering 719, 720, 728, 737, 764, 772, 779 Interstitially ordered superstructures 764 Interstitials 231, 232, 236, 339, 382, 561, 765 Intraplanar ordering 767 Intrinsic diffusivities 571 Invariant line strain 276, 555, 559, 614, 642, 643, 796 Invariant line 276, 302, 342, 504, 555, 559, 614, 642, 643, 644, 645, 646, 648, 650, 655, 663, 679, 796, 798 Invariant plane strain (IPS) 45, 47, 110, 260, 266, 504, 632, 728, 737, 796 Invariant plane strain condition 45, 110, 504, 645, 734, 737, 779 Invariant plane strain transformation 45, 47, 276, 632, 651, 732, 734, 779 Invariant plane 45, 47, 110, 260, 267, 276, 314, 504, 532, 644, 651, 654, 728, 732, 734 IPS condition 270, 272, 275, 291, 294, 304, 315, 643, 662, 665, 733, 735, 736 Irrational 244, 246, 267, 333, 347, 633, 648–651, 732 Isomorphous 22, 26–30, 282, 285, 287, 327, 418, 475, 586–588, 606, 609, 616, 620, 765 Isothermal kinetics 194, 196, 202, 264 Kagome net or K net 55 Kirkendall effect 572, 585

806 L12

54, 55, 59, 61, 201, 228, 382, 390, 391, 392,402, 408, 409, 413, 436, 439, 440, 441, 442, 529, 530, 533 L1o (tP4, AuCu type) 54, 59, 61 La78 Ni22 163 Lamellar 111, 382, 420, 440, 441, 443, 444, 448, 450, 451, 452, 453, 454, 456, 457, 629, 635, 671, 673, 675, 676, 679, 681, 682, 688, 691, 696, 697 Lamellar eutectoid 635 Lamellar microstructure 450, 451, 691 Lamellar structure 443, 450, 456, 457, 673, 675, 676, 681, 682, 697 Lath martensite 48, 308, 310, 320, 328, 329, 331, 332, 437 Lath morphology 48, 277, 310 Lattice collapse mechanism 471, 474, 492, 498, 499, 504, 509, 518, 527 Lattice correspondence 45, 103, 289, 305, 342, 363, 368, 431, 485, 488, 515, 786 Lattice distortion matrix 731 Lattice-invariant shear (LIS) 45, 260, 269, 271, 293, 295, 296, 312, 314, 315, 346, 347, 366, 368, 504, 651, 732, 733, 779, 796 Lattice parameters 7, 46, 51, 227, 290, 345, 427, 443, 512, 520, 602, 613, 644, 723, 786, 798 Lattice shear 44, 46, 120, 270, 271, 345, 532, 632, 737, 777 Lattice-site correspondence 560 Lattice strain dominated 111 Lattice strains 45, 46, 100, 106, 111, 269, 270, 271, 431, 663, 731, 786 Laves phase 55, 59, 60, 62, 235, 250, 251, 622, 658, 754, 756, 759, 761 Linear muffin tin orbital (LMTO) 9, 11, 15, 19, 385, 412 Liquid/solid interface 129, 134, 135, 146 Local density approximation (LDA) 11, 14, 15, 16, 385, 386 Localized hydride-embrittlement 746 Long-range order (LRO) 387 Mackay cluster 251, 252 Macroscopic flow behaviour 335 Macroscopic shape deformation 109

Index Macroscopic shape distortion 300, 301 Macroscopic shear 44, 266, 270, 274, 298, 303, 372, 504 Macroscopic strain 264, 270, 275, 504, 547, 704, 734 Macrosegregation 141, 143–145 Martensite interfaces 264, 280, 680 Martensite 26, 30–32, 35, 43–49, 106, 110–112, 153, 155, 260, 261, 263–267, 269–271, 273–281, 283, 287, 290, 293, 295, 296, 302–310, 313–317, 319–329, 331–340, 347–352, 354, 355, 358–362, 365, 368, 370, 372, 418, 421, 437, 504, 532, 546, 559, 609–618, 626, 635, 640, 645, 651, 666, 680, 720, 732, 735, 753 Martensite phase 26, 43, 44, 48, 279, 280, 305, 354, 355, 357, 421, 618 Martensitic transformation 25, 26, 43–45, 47, 49, 100, 101, 105, 106, 109–111, 113, 114, 155, 260–266, 270, 275–279, 281–285, 287, 288, 290, 292, 293, 306, 312, 324, 326, 327, 339, 342, 348, 352, 354–356, 370–372, 418, 419, 447, 475, 492, 494, 500, 504, 530, 623, 627, 632, 642–644, 646, 651, 653, 663, 777–779 Martensitic 25, 26, 34, 35, 38, 43–47, 49, 64, 100, 101, 105, 106, 109–114, 141, 148, 149, 149, 153–155, 260–266, 270, 275–285, 287, 288, 290, 292, 293, 297, 302, 304, 306, 312, 319, 320, 322, 324, 326–328, 331, 337–340, 342, 348, 350, 352–356, 361–363, 368, 370–372, 418, 421, 439, 447, 475, 477, 482, 492, 494, 500, 504, 530, 546, 559, 609, 623, 626, 627, 632, 642, 643, 644, 651, 653, 658, 662, 663, 668, 704, 729, 736, 738, 769, 777–779 Massive 48, 106, 109, 110, 185, 205, 217, 221, 223, 281, 421, 447, 448, 450, 453–456, 559, 623–626, 628–630, 671 Massive transformation 106, 109, 110, 185, 205, 421, 447, 448, 450, 453–456, 559, 623–626, 628–630 Matano interface 574 Maximum resolved shear stress 303 Mean field 94, 385, 388

Index Mean free slip length 332, 334, 335, 337 Mechanically driven systems 226 Mechanism 101–104, 228, 451–453, 499–518, 560–562, 589, 648, 737 Melt extraction 150 Melt spinning 150, 151, 173, 191 Metallic glasses 110, 134, 157, 158, 162, 164, 165, 171, 176–179, 181, 183–185, 187, 193, 205, 207, 210 Metal–metal glasses 181, 187, 188, 204 Metal–metalloid glasses 187, 204 Metastable 23, 35, 53, 92, 129, 215, 403, 410, 502, 611, 671, 793 Metastable Zr3 Al 337, 382, 437, 441, 530 Mf 47, 48, 110, 278, 283, 350, 353–356, 360, 361, 363 Microsegregation 141–144 Microstructural evolution 235, 555, 603, 656, 683, 684, 685, 691, 697 Minor orientations 314 Mirror plane 272, 274, 275, 293, 296, 314, 315, 342, 346, 351, 356, 614, 679 Miscibility gap 29, 31, 35, 52, 54, 98, 99, 409, 410, 555,558, 588, 589, 596, 597, 598, 599, 601, 609, 621, 622, 787 Misfit compensating ledges 650 Mixed diffusive/displacive transformation 120 Mixed Mode Transformations 115 Modes of crystallization 185, 200 Molar free energy 131, 285, 435, 571, 589, 598, 625 Molecular dynamics (MD) 16, 235, 385 Monoclinic 14, 41, 54, 62–67, 113, 340, 345, 347, 348, 362–366, 369–372, 770 Monotectoid reaction 29, 31, 32, 35, 53, 187, 555, 559, 587, 606, 607, 609, 621, 787 Monotropic transitions 92 Monovacancies 561, 562 Monte Carlo (MC) 14, 16 Morphological stability 135, 138, 139 Morphologies 48, 51, 172, 173, 277, 304, 309, 343, 500, 626, 633, 638, 640 Morphology and substructure 48, 304

807 Ms

44, 47, 48, 101, 110, 112, 113, 140, 154, 155, 262, 263, 265, 277, 278, 281–283, 287, 288, 303, 310, 312, 313, 320, 322, 324, 325, 331, 361, 363, 418, 627 Muffin Tin (MT) 15 Multilayered structures 237 m-ZrO2 63

Nearly-free-electron (NFE) 18 Neutron irradiation 235, 237, 702 Ni10 Zr7 578 Ni5 Zr 578 Ni5 Zr2 579 Ni–B 162, 187 NiZr 578 NiZr2 578 Nodular corrosion 771 Non-ferrous 260, 326 Non-collinear 292 Non-Equilibrium Phases 23, 26, 43 Nucleation 97, 98, 101, 105, 109, 110, 113, 116, 120, 122, 146, 158, 162, 164, 165, 167–172, 175, 185, 188, 192–194, 196, 197, 199, 200, 202, 206, 208, 211–213, 217, 222, 223, 225, 226, 229, 265, 277–279, 305, 322, 324, 325, 348, 349, 381, 439, 451, 452, 455, 479, 494, 527, 528, 533, 539, 547, 580, 582, 589, 592, 594–597, 612–614, 618, 620–623, 625, 629, 630, 633, 640, 657, 661, 666–668, 670, 671, 674, 683, 694, 698, 726, 739, 740, 751, 771 O-phase 417, 418, 420, 421, 422, 423, 427, 428, 431, 433, 434, 435, 555, 556, 662, 665, 666, 786 Octahedral and tetrahedral voids 57 Omega phase 49 Order–disorder 235, 380–382, 384, 386, 388, 389, 401, 414, 459, 460, 767 Order–disorder transformation(s) 235, 384, 386, 767, Ordered -Structures 421, 518–536 Ordering tie lines 458

808 Orientation relations 45, 50, 103, 175, 192, 267, 293, 302, 303, 310, 313, 314, 363, 443, 454, 488, 613, 635, 642, 679, 798, 799 Orientation relationship 45–47, 50, 51, 103, 175, 192, 274, 276, 293, 302, 303, 310, 362, 364, 365, 368, 383, 420, 424, 440, 441, 444, 457, 485, 530, 629, 632, 633, 641–644, 677, 730, 735, 737, 738, 741 Orientational relation 778 Orientational variant 506 Orthohexagonal 281, 290, 296, 532, 663 Orthorhombic 46, 54, 66, 111, 113, 200, 282, 345, 418, 427 Orthorhombic -martensite 281, 283, 418, 421, 616–618 Oxidation 383, 443, 769, 770, 772 Paraelectric 113 Partial dislocations 279, 316, 452, 453, 455 Partial molar volume of hydrogen 719, 727 Partially crystalline alloys 171 Partially Stabilized Zirconia (PSZ) 65, 369–371, 373 Partitionless polymorphic solidification 161 Partitionless solidification 110, 131, 133–135, 156, 160, 161, 185, 205 Peierls–Nabarro 339 Peritectoid reaction 27, 141, 187, 382, 433, 434, 437, 441, 442, 529, 772 Perovskite structure 113 Phase diagrams 7, 8, 11, 13–15, 17, 18, 24, 26–29, 43, 53, 54, 65, 98, 129, 133, 134, 160, 161, 163, 218, 232, 282, 286–288, 369, 387, 388, 410, 413, 414, 418, 475, 479, 516, 587, 588, 606, 609, 624, 626, 722, 764, 765 Phase rule 7, 92 Phase separation 43, 52, 53, 99, 115, 185, 210, 327, 409, 411, 500, 523, 555, 559, 587, 603, 606, 609, 612, 616, 618, 620, 621, 787 Phase separation in -phase 43, 52, 53, 559, 587, 589, 603, 606, 609, 618, 620–622

Index Phase stability 13, 14, 16–18, 24, 27, 52, 62, 148, 237, 240, 284, 383, 384, 433, 508 Phase transformation (transition) 5, 8, 21, 62, 89–92, 94, 105–107, 109, 115, 120, 128, 140, 145, 147, 148, 152, 153, 157, 176, 186, 195, 271, 285, 352, 361, 369, 382, 383, 386, 409, 417–419, 430, 443, 450, 474, 495, 512, 529, 530, 535, 558, 560, 606, 607, 609, 619, 621, 623, 643, 656, 684, 706, 721, 725, 726, 753, 754, 769, 772 Phenomenological theory of martensite crystallography 266, 270, 504, 720, 735, 729 Phonon dispersion curves 498, 507 PHWR 382, 706, 707, 749, 772 Pilger milling 702 Plasma rotating electrode 150 Plastic accommodation 313, 331, 347, 719 Plastic deformation 178, 263, 265, 324, 325, 334, 354–356, 561, 683, 684, 688, 691, 702–704, 726 Plate morphology 49, 277, 309, 313, 328, 644, 729 Plate-shaped 504–506 Plateau pressures 761 Point defects 231, 232, 233, 235, 533, 561 Polarized light microscopy 302 Polydomain 277, 305, 323, 350 Polymorphic crystallization 110, 175, 178, 185, 187, 189, 191, 194, 197, 202, 204, 205 Polymorphic transformations 65, 92, 110, 223, 362 Polymorphism 4, 5 Polymorphous transformation 5 Portevin–Le Chatelier (PLC) effect 539 Post-solidification transformations 140 precipitate particles 370, 659 Premartensitic 279, 361 Pressure-induced → 494 Pressure–composition isotherms (PCI) 756, 758, 761 Primary crystallization 177, 187–189, 192–194, 196, 202–204 Primary plates 304, 313, 348, 349 Principal axes of deformation 268 Principal strains 46, 268, 301, 366, 662, 778 Processing maps 699

Index Promotion energy factor 17 Pseudoelasticity 339, 340 Pseudoplastic 353–355, 359 Pt–Sb 162 Quantum Monte Carlo (QMC) 14, 16 Quantum structural diagrams (QSD) 17 Quasicrystalline structures 241, 243 Quenched-in nuclei 165, 172, 192, 197, 199, 203 R phases 103, 342 Radiation-induced amorphization 229, 235 Rapid or nonquenchable 105 Rapid solidification 36, 128, 139, 150, 152, 153, 155, 159, 160, 169–171, 228, 231, 250, 522 Rapid solidification processing 36, 150–153 Recovery 346, 353, 684, 692, 710 Reconstructive transformations 104 Recrystallization 145, 199, 421, 670, 683, 684, 692, 710 Resistivity 181, 279, 340, 341, 362, 479, 482, 768 Reversion stress 260, 356, 359 Rigid body rotation 45, 103, 271, 274, 276, 292, 314, 635, 643, 644, 663, 704, 705, 736 Rupture 371, 382, 771 Schiebung 279 Schmidt factors 334 S–d electron transfer 7, 11, 12, 20 Second (or higher order) 5, 502 Second-order transitions 93, 115, 528 Self-diffusion 176, 564, 568, 604, 688, 689, 690, 728, 729 Self-similarity 241, 313 Self-accommodation 277, 305, 306, 308, 320–323, 340, 342, 345, 347, 349, 350, 355, 361, 362, 666, 668, 669 Serrated flow 540, 542, 545, 546, 550 Shape memory effect 339, 340, 352, 353, 355, 356 Shape strain 263, 265, 269–271, 303, 306, 308, 316, 320, 322, 366, 373, 504, 644, 666, 668,733, 734

809 Shear band 693, 694 Shear direction 103, 295, 296, 302, 303, 506, 733, 734 Shear instability 213, 215 Shear moduli 337 Shear modulus 25, 47, 112, 215, 260, 331, 337, 538, 720 Shear plane 295, 296, 504, 733 Shear transformation 111, 120, 721, 728, 729, 737, 798 Shock pressure 471, 481, 489, 499, 500, 504 Shock wave 8–10, 506 Shuffle dominated 111 Similarity transformation 272, 294, 732 Single surface trace analysis 310 Sintering 370 Site Occupancies 458, 520 Size factor 17, 23, 29, 338 Slip mode 322, 334 Sluggish or quenchable 105 Softmode 111 Solid solubility 27, 28, 65, 150, 152, 285, 573, 722, 725–727, 746, 747 Solid State Amorphization 212, 215, 226, 228, 229 Solidification 36, 110, 128–142, 144–148, 150–156, 159–161, 163, 169–171, 175, 185, 205, 206, 228, 231, 250, 522, 690 Solubility limit 66, 152, 327, 609, 629, 721, 722, 746 Solute partitioning 121, 131, 175, 492, 629, 637, 670, 681 Sp-band 18 Special point ordering 401 Specific volume of hydride precipitate 727 Spinodal clustering 98, 99, 115, 116, 409, 438, 592 Spinodal decomposition 99, 156, 524, 525, 559, 589, 592, 593, 595, 597, 603, 604, 616, 617, 618, 785 Spinodal ordering 98, 99, 403 Splat quenching 150, 151, 175 Square, S net 55 Stacking domains 319 Static concentration wave 389 Static concentration wave model 389 Stereographic projection 316, 644, 732, 736

810 Stiffness moduli, C11 , C44 and C12 25 Strain coupling 305, 306, 355 Strain energy density 166, 306, 307, 308, 349, 719, 725 Strain energy 47, 99, 166, 238, 265, 277, 305, 347, 370, 479, 504, 592, 666, 719, 724, 727 Strain hardening 683 Strain-induced martensitic 324 Strain rate sensitivity 540, 548, 684, 685, 693, 696 Strain spinodal 279, 603 Strain-induced plates 265 Strengthening mechanisms 327 Strengthening 22, 326–328, 331, 338, 698 Stress-assisted 48, 265, 322, 324–326, 354, 546 Stress-assisted martensite 260, 324–326 Stress directed migration 744, 745 Stress free transformation strain 259, 306, 719, 727 Stress reorientation of hydrides 742, 743 Stress–strain 325, 335, 337, 352–354, 356, 358, 537, 540, 547, 548, 550, 684, 687, 690, 692, 698 Structural ledges 633, 649, 650, 652, 796, 796, 798 Structural relaxation 158, 176, 180, 181, 183, 184, 193, 412 Structure maps 17 Substitutional 23, 27, 30, 177, 277, 327, 338, 339, 387, 415, 533, 539, 546, 559, 561, 562, 609 Substructure 45, 48, 49, 293, 303–305, 313, 316, 317, 319–322, 327–329, 332, 421, 626, 670, 687, 689 Supercooled liquid 137, 140, 160, 163, 164, 182, 207, 208, 211–213 Supercooling 47, 99, 129, 133, 137, 138, 154, 161, 213, 214, 262, 263, 265, 281, 287, 324, 361, 403, 416, 438, 479, 527–529, 623, 638, 658 Superlattice 99, 109, 117, 227, 320, 380, 381, 419, 437, 451, 454, 458, 518, 527, 679, 720 Superplasticity 685, 691, 695, 701

Index Supersaturation 120–122, 236, 282, 283, 327, 438, 441, 447, 453, 457, 609, 619, 654, 658 Surface relief 266, 269, 270, 310, 362, 364, 492, 632, 737, 769 Symmetry 5, 6, 15, 51, 52, 55, 58, 99–101, 114, 117, 176, 185, 214, 241, 243–245, 251, 271, 272, 305, 316, 370, 386, 396, 401, 403, 406, 416, 421, 422, 424–429, 431, 450, 485, 495, 496, 500, 502, 519, 520 Symmetry tree 471, 534, 535 Sympathetic nucleation 655 Syncretist Classification 105 TEM 45, 49, 173, 189, 191, 196, 220, 250, 310, 331, 451, 648, 675, 692, 789 Tempered 284 Tempering 284, 327, 421, 559, 609–613, 615–618 Tempering of martensite 555, 559, 609, 615 Temporary Alloying 753 Tensile 145, 157, 270, 290, 291, 335, 352, 356, 370, 372, 540, 689, 692, 696–698, 707, 708, 710, 727, 741–746 Terminal solid solubility 717, 719, 725, 726 Terminal solid solubility for precipitation (TSSP) 725, 726 Terminal solubility 737, 741 Tetragonal 12, 54, 55, 63, 65, 66, 113, 362–368, 372 Tetragonal phase (t-ZrO2 63 Tetragonal shear constant 12 Texture 691, 701, 702, 704, 707, 743, 745, 769 The C11b (tI6, MoSi2 type) structure 60 Thermal arrest memory effect 340, 360, 361 Thermal cycles 354 Thermal expansion coefficient 369 Thermal migration 744, 745 Thermal shock resistance 369 Thermal stability 176, 181, 208, 210 Thermo-Mechanical Processing 683, 689, 690 Thermochemical processing 753, 754 Thermodynamic interaction parameter 26

Index Thermodynamics 8, 92, 93, 128, 129, 133, 215, 220, 261, 288, 386, 387, 393, 408, 609, 619, 623, 685, 756 Thermoelastic equilibrium 265, 340, 354, 356, 358, 359 Thermoelastic 265, 340, 350, 354–356, 358, 359 Thermomechanical processing (TMP) 684 Thin Film Multilayers 237 Thin twins 314, 315 Threshold stress 356, 548, 550, 745, 747 Threshold value of stress 745 Ti-V 11, 27, 29, 30, 32, 33, 53, 288, 325, 482, 483, 489, 606, 609 Ti2 AlNb 421, 422, 423, 432, 433, 670 Ti2 Al–Nb 669 Ti2 Cu 662, 676, 677 Ti2 N 39, 721, 772 Ti45 Zr38 Ni17 251 Ti4 AlNb3 669, 670 Ti5 Al3 520 Ti–Ag 576, 578, 624, 626 Ti–Al 27, 29, 30, 34, 35, 284, 377, 381, 383, 412, 413, 414, 415, 416, 430, 433, 446, 453, 464 Ti–Al–Nb 383, 417, 421, 430, 432, 433, 519, 662, 669 Ti-Al-V 284 Ti–Au 383, 624, 626, 627 Ti–Be 160, 161, 188 Ti–Bi 673 Ti–Co 673, 674 Ti–Cr 27, 29, 30, 33, 34, 250, 316, 474, 621, 622, 638, 639, 640, 648, 651, 673, 758, 798 Ti-Cr-Si 250 Ti–Cu 27, 157, 160, 163, 179, 232, 310, 673, 674, 675, 676, 677, 682 Ti–Fe 161, 582, 583, 586, 626, 673, 674 Tight-binding (TB) 18, 19, 215 Ti–H 27, 717, 719, 722, 723, 728 Ti–Mn 27, 250, 314, 316, 673, 758 Ti-Mn-Si 250 Ti–Mo phase diagram 621 Ti–Mo 27, 29–32, 53, 288, 536, 537, 546, 606, 609, 616, 621

811 Ti–N 27, 29, 30, 39, 220, 232, 339, 340, 458, 464, 477, 482, 606, 607, 620, 673, 772, 774, 776–778 TiN 39, 40, 721, 772, 778 Ti–Ni Shape Memory Alloys 339 Ti–Ni 27, 220, 232, 339, 340, 673 Ti–O 27, 764, 765, 767, 769 Ti–Pb 673 Ti–Pd 356, 673 Ti–Pt 673 Ti–Si 624, 626, 627, 661 Titanium (Ti) 4, 7, 9, 21, 24, 626, 629, 656, 687, 690, 691, 698, 706, 720, 721, 723, 737, 738, 744, 753, 754, 769, 772, 777 Titanium nitride 721, 772 Ti–V 27, 29, 30, 32, 33, 53, 288, 325, 482, 483, 489, 606, 609 Ti–X 24, 26–29, 43–46, 48, 53–55, 57, 59, 61, 62, 148, 418, 657, 671, 673–676 Ti-Zr 27, 29, 30, 249–252, 285–287, 327, 758 Ti-Zr-Fe 249–251 Ti-Zr-Ni 249, 251, 252 Topologically close packed (TCP) structures 57 Toughening 369–372 Toughness 22, 65, 145, 327, 369–371, 655, 706, 742, 747 Tracer-diffusion 555, 564, 566 Transformation-induced plasticity 371 Transformation sequences 141, 257, 340, 377, 409, 428, 429, 430, 432, 447, 530, 621, 786 Transformation temperatures 47, 323, 340, 729 Transformation twins 49, 334 Transient nucleation 168–170 Transition metals 4, 7, 8, 15, 18, 20–22, 24–26, 29, 157, 162, 163, 177, 225, 231, 249, 480, 507, 516, 518 Transmission electron microscopic 737 Triangular, T net 55 TRIP steels 371 Triple point 8–10, 218, 219, 480 True plastic strain 260, 328 True strain 328, 696 True stress 328, 337

812

Index

TSS 719, 720, 725, 742, 745, 751 TSSD 725, 726, 727, 751, 752 TSSP 726, 727, 751, 752 Twinning 270, 271, 274, 275, 277, 295, 296, 300, 313, 322, 333, 342, 346, 355, 445, 484, 505, 546, 702, 728, 729, 737, 798 Two-way shape memory 354 Type I twins 296, 345, 346, 347, 350, 351, 730, 735, 736, 738, 739 Type II twins 346, 347, 351 Umklapp 279 Undistorted 45, 46, 267–271, 274, 292, 293, 295, 296, 301, 485, 505, 509, 644, 664, 732, 736 Unrotated 45, 241, 267, 271, 274, 300, 614 Vacancy mechanism 561, 562, 564, 565, 566, 569, 571 Vapour pressure 756, 759, 760, 761, 764 Variant 146, 351, 527 Viscosity 157, 158, 161, 165, 168, 170, 171, 175, 177, 184, 206, 210 Vitrification 128, 157, 158, 160, 164, 165, 171, 172, 175, 192–194, 208, 215, 221–223 Wechsler, Liebermann and Read (WLR) methodology 732, 734, 735 Weldments 145 Widmanstatten 49, 141, 148, 149, 327, 328, 337, 338, 421, 435, 447, 450, 610, 635, 648, 674, 689, 692, 693 Widmanstatten side plates 635 WLR 732, 734, 735 WLR theory 734 X-ray diffraction 739 Yield strength

165, 362, 363, 474, 482,

324, 339, 537, 655, 745

Zener anisotropy ratio 25–26 Zigzag habit 314 Zircaloys 23, 30, 658, 659, 661 Zirconia 62, 63, 65, 227, 362, 365, 366, 368–371, 770

Zirconium (Zr) 4, 7, 10, 23, 24, 656, 661, 701, 702, 706, 720, 721, 727, 737, 739, 741–745, 747, 751, 758, 769–772 Zr alloys 4, 23, 24, 30, 47, 111, 120, 128, 145, 150, 152, 171, 172, 283, 284, 287, 313, 320, 327, 338, 381, 475, 501, 525, 587, 604, 658, 673, 683, 691, 692, 701 Zr-Nb-Sn-Fe 661 Zr(Cr,Fe)2 658, 659 Zr(Nb,Fe)2 661 Zr2 (Fe,Ni) 658, 659 Zr–2.5% Nb 24, 294, 312, 613 Zr2 Al 38, 39, 141, 156, 210, 382, 437, 441, 442, 518, 519, 525, 527–530, 584 Zr2 Al, Ti2 Al 518 Zr2 Cu 211, 212, 222, 676, 677, 679, 681, 682 Zr2 Ni 174, 175, 188, 189, 200, 202, 204 Zr3 (Fe, Ni) 190 Zr3 Al 30, 38, 39, 55, 61, 227, 228, 382, 436,439, 440, 441, 442, 523, 525, 529, 530, 531 Zr3 Fe 35, 191, 192, 200, 202, 681, 682 Zr4 Fe 681 Zr5 Al3 38, 141, 520, 521, 525, 527, 528, 529 Zr5 Al4 521, 522 Zr76 Fe12 Ni12 200 Zr76 Fe16 Ni8 170–172, 200 Zr76 Fe20 Ni4 200, 204 Zr76 Fe24 189, 191, 194, 200, 202 Zr–Al 29, 30, 38, 39, 116, 140, 156, 207, 208, 210, 381, 382, 437, 522, 523, 530, 574, 582, 584 Zr–Co 163, 220 Zr–Cr 23, 658, 758 ZrCr2 62, 658, 758, 760, 761 Zr–Cu 157, 160, 163, 207, 218, 220, 222, 223, 232, 556, 675, 676, 681, 682 Zr–Fe 29, 30, 35, 36, 163, 172, 187, 195, 200, 220, 232, 249, 250, 251, 676, 681, 682 Zr–H 29, 30, 39, 41, 717, 722, 723, 724, 728, 740 Zr–Nb 11, 23, 29, 30, 35, 52, 53, 153, 154, 155, 310, 477, 481, 492, 515, 586, 599, 600, 601, 603, 606, 607, 609, 618, 620, 621, 640, 645, 649, 650, 661, 701, 798

Index (Zr,Nb)3 Fe 661 Zr–Nb system 30, 35, 599, 601, 603, 607, 609, 618 Zr–Ni 157, 160, 163, 164, 172, 200, 204, 218, 220, 221, 225, 232, 249, 251, 252, 578, 579, 675 Zr–Ni–Cu 160 Zr–Ni–Fe 163 Zr–O 29, 30, 41, 63, 764–767 ZrO2 polymorphs 62, 63

813 ZrO2 62, 63, 65, 66, 363, 770 ZrO2 –CaO system 66 ZrO2 –MgO system 66 ZrO2 –Y2 O3 system 67 Zr–Sn 29, 30, 36, 38, 658 Zr–Ta system 52, 53, 602 Zr–Ti 310, 328, 334–338, 586, 692 Zr–X 24, 26, 27, 29, 43–45, 48, 53–55, 57, 59, 61, 62, 148, 657, 673, 675, 676


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